Muir on research in Scotland

Thomas Muir was the second president of the Edinburgh Mathematical Society, being elected in 1883, the year in which the Society was founded. He gave his presidential address on The Promotion of Research with Special Reference to the Present State of the Scottish Universities and Secondary Schools on 8 February 1884. It was intended that the address be unpublished, but members of the Society persuaded Muir to have it printed. He explains in the Preface that he would have liked to have produced a more substantial paper on the topic for publication, but did not have the time to rewrite the address. Eventually the address appeared as an addendum to the second part of the Proceeding of the Edinburgh Mathematical Society. It is interesting that the address was given before the Society decided to publish a Proceedings and Muir argues strongly in favour of a Proceedings in his address.

Special Reference to the Present State

(An Address delivered before the Mathematical Society of Edinburgh, 8th February, 1884.)


Privately Printed.


This address, written hurriedly during the pressure of other work, was undertaken in order to discharge an obligation to the Mathematical Society, and was not meant for further publication. Immediately after it had been read, however, a strong desire was expressed by office-bearers and other members present to have it printed and published in the ordinary way. If I had from the first intended it so to appear, I should doubtless have adopted a somewhat different style, and have taken pains to make it a more substantial contribution to the discussion of so important a subject. As I cannot afford the time now to recast it, I print it with some hesitation exactly in the form in which it was read, hoping merely that it may find readers as indulgent as the Edinburgh audience for which it was prepared.


Mr Chairman and Gentlemen,

Immediately after you had done me the honour to elect me President of the Society, I somewhat unexpectedly learned that in the programme of the session there was included a President's address. In ordinary circumstances it would have been more pleasing to me to have the item entitled the President's Paper, or the President's Half-hour at the Black Board. As matters stand, however, the society being still young in years, and still engaged in feeling its way towards firm ground, it seems to me there are not a few things worthy of our consideration, which can be treated of without the use of chalk and symbols.

It is of importance to us, for example, to recall at intervals the ends we had in view in uniting ourselves together to have suggestions made of any new means whereby these ends may be the better furthered, and to take interest, as a body, by discussion and friendly criticism, in anything outside the society that either seeks the same ends as we do, or exercises an unfavourable influence on the advancement of them. Again, it is of value to us to hear, as we did last session, some one who has travelled in more realms than one of the mathematical universe, tell us in what districts labour and skill are urgently needed, what lands are lands of promise, and what kind of workmen this or that field is suitable for.

Of these two classes of subjects I select the former as being that with which I am the less unfitted to deal in the so-called address. I say 'so-called' because I have a lingering feeling against the use of the term. For the nonce I should wish separated from it the ideas of authoritative wisdom and paternal advice which it sometimes connotes. In what I have got to say I have no counsel to give the members of the society: my wish is rather to take counsel with them on a subject in which, as their membership implies, they evince a cordial interest - the Promotion of Mathematical Research.

I assert at the outset as an established fact that we do not, as a nation, take that part in research which, considering the amount of our population, our advancement in civilization, and the acknowledged capabilities of our people, we ought to take. Writers of other nationalities tell us so, and many of our own most eminent men have thought it their duty to draw attention to the fact. And what is true of research generally is more manifestly true of research in pure Mathematics. Whatever means of publication we think of - books, journals, or the serials of learned societies, - we find they have all the same tale to tell of disproportionate unproductiveness. Books containing original work in pure Mathematics are now rare in almost any country; in Scotland, however, their number is practically nil: of journals, we have none: and our only society to be honourably mentioned as publishing memoirs and shorter communications on the subject of Mathematics is the Royal Society of Edinburgh. It is the Transactions and Proceedings of this Society alone, that have for a long period of years saved us from utter reproach. (We have good ground to hope that as it has been in the past with the Royal Society, so it will be even more abundantly in the future: and such, I am sure, is the sincere desire of every member of the Mathematical Society; not only so, but we should even consider that we had fulfilled part of our aims, if our Society were the means of encouraging and bringing to light young men of ability in research, who might honourably aspire to recruit the ranks of the Royal Society.) One is, of course, reminded in stating facts of this kind regarding the backwardness of Scotland, that the home of Scotsmen is abroad, and that, were we to take nationality as against nationality, the result might be different. No doubt there are crumbs of comfort to be got in this way. They do not, however, by any means make up the deficiency; and even if they did, the serious question would remain, whether it was good or bad policy for a country to export more than it could well spare. But let any one try to count up the number of Scotsmen resident in England or abroad who publish original work in pure mathematics, and the stern fact will meet him as before, that though we may produce many promising students of mathematics and our share of successful examinees, we do not turn out mathematicians able and willing to advance the boundaries of their subject.

Little need be said by way of pointing out that this is not as it should be. Workers in even the nonmathematical sciences would consider it a matter for regret, and students of general history would regard it as a fact of unpleasing import. The reason for it is not that Mathematics is despised or underrated in Scotland: we know, on the contrary, that it is as widely taught here, and valued at the very least as highly as in any other country. Nor can this so far gratifying fact be for a moment held as a palliation of the other: it rather only adds increased significance to it. It points to teachers and methods of teaching that are either devoid of vitality, or are alive with a depressing utilitarianism. I venture to say that the teacher, who never feels a desire to know more of his subject than what his text-books give him, takes a low view of his profession; and that the teaching, which never kindles in the hearer a spark of enthusiasm for his subject, or awakens in him a longing to know more and to explore for himself, is not the highest kind of teaching, but fails in a vital point. As for the historical significance of the decline of original research in a nation, so far as this regards Mathematics I cannot do better than quote the eloquent and impressive words of the late Professor Henry Smith of Oxford, who from his wide culture and general robustness of mind was one of the last men to take a one-sided or pessimistic view. "In these days," he says, "when so much is said of original research and of the advancement of scientific knowledge, I feel that it is our business to see, that so far as our country is concerned, mathematical science should still be in the vanguard of progress. I should not wish to use words which may seem to reach too far, but I often find the conviction forced upon me that the increase of mathematical knowledge is a necessary condition for the advancement of science, and, if so, a no less necessary condition for the advancement of mankind. ... Perhaps also," he continues, "it might not be impossible to show, and even from instances within our own time, that a decline in the mathematical productiveness of a people implies a decline in intellectual force along the whole line; and it might not be absurd to contend that on this ground the maintenance of a high standard of mathematical attainment among the scientific men of a country is an object of almost national concern."

When we begin to think of the agencies whereby mathematical research may be revived, the mind turns at once to the UNIVERSITIES, one of the great functions of which is the advancement of knowledge. Of recent years, it is true, doubts have been expressed as to the wisdom of combining instruction and research, and strong arguments have been given for the establishment of national institutions devoted to research alone. Colonel Strange's opinion on this point was very strong, and no man of his time did more than he to draw attention to the whole subject, and to rouse the country from the languor into which it had fallen. In his evidence before the Royal Commission on the Advancement of Science, he says:- "I consider education to be quite a different thing from national research, that they should be kept as distinct as possible, and that one great evil now existing is the mixing up of those two things." Although Colonel Strange was not alone in holding this view, the great bulk of opinion, now at least, is in favour of it only so far as certain special subjects are concerned, the line being drawn so as to include such departments of science as Astronomy, Meteorology, Solar Physics, etc., which more immediately affect the national welfare, and for the pursuit of which extensive collections of apparatus are required at particular spots of the earth's surface. At any rate, public opinion is not ripe for the separate endowment of Mathematical research;- and yet, had the Government stepped forward when it was necessary for Professor Sylvester to leave his country and go to America, and said, "We shall be proud to have you draw upon us for a sufficient income: do as you have been doing, and we shall ask no questions," - had the Government, I say, done this, there would have been many to applaud and few but the ignorant to condemn.

To the Universities, then, in the first place, we must look for aid. Leaving for the moment the question of the doing of original work by the professors themselves, let us look at the means employed for making original workers out of promising students. It is in this we in Scotland most fail. We recognise two of the functions of a University - instruction and research; we ignore, so far as mathematics is concerned, a third and equally important function, - instruction in research. A Scotch University student who has a special taste for mathematics, and has come to the University to develop that taste, has usually something like the following career:- Of the two or three mathematical classes taught in the University, he very probably enters the highest. There he obtains a knowledge of Synthetic and Analytical Conics, the elements of the Differential Calculus, and, it may be, of the Integral Calculus as well. He knows there is no hope for him if he does not take his Master of Arts degree, and he gives his attention to Classics and Mental Philosophy with this end in view, continuing by himself his reading in Mathematics as far as it may be possible to do so. In time he graduates: this entitles him to compete for a scholarship: he competes, and is successful, leaves for Cambridge, and his University knows him no more. Probably in the newspapers we observe that Mr Donald Scott of a certain northern university has gained an open scholarship at Johnshouse, and the competition having been between him and a number of young men fresh from the English public schools, we are gratified accordingly with his startling success. Gentlemen, I put it to you, if this is a thing for us as Scotsmen to be altogether proud of. When in these cases a young Scotch student competing with English students of the same age gains a scholarship, there may be cause for gratulation: but the Scotsman who glories in the part his Universities play in the matter glories in his own shame. Is it really past hoping for, that all this may yet be changed? Is it altogether absurd to suggest that a graduate who gains a scholarship should remain during the tenure of his scholarship in his own university, there to grow in knowledge under his favourite professor's guidance, to learn to teach, to be initiated into independent research, and to be a cord of strength to his alma mater and to his country? Surely there are no vested interests with ogre eyes in the way here. There need be no additional lectures for the scholar's benefit:- if at the age he has reached he cannot pursue his own reading in the higher subjects, it may be safely affirmed that he never will. His experiments in teaching need be a loss to no one:- to many a student preparing for a degree examination, or desirous of knowing a special subject, they would undoubtedly prove a gain. His attempts at independent literary work need harm nobody:- his professors would be only too glad to put minor problems in his way, and if by chance he had the enthusiasm to carry him safely through a piece of the so much wanted bibliographical work, fellow-labourers the world over would rise and call him blessed.

It may be objected to this scheme, that the independent literary or scientific work of such an author would not be worth much. I reply that the scheme is not proposed with this intention. The work would be avowedly 'prentice-work, and the time to judge of its value would come when the author had reached the full maturity of his powers. But I go farther, and say, that although it is not a sine qua non that the work should be valuable, still it is absolutely certain that not a jot of it need be valueless, or unworthy of being put on record. The amount of useful skilled labour which is needed for the advancement of mathematics, or indeed of any science, is practically unlimited; and surely a graduate capable of gaining one of the few scholarships or fellowships of a poor but numerously-attended university, is capable of at least this skilled labour. And then, who doubts but that what has happened elsewhere would happen with us, that occasionally men of original power would arise whose very first-fruits would be all-important, and who would redeem in one year more years of mediocrity than their predecessors had left to be redeemed.

But next the questions may be asked - What careers are there open for such men after they have completed their post-graduate course? Is there anything like the same possibilities for them as are within the reach of Cambridge wranglers? The answer to the first question is, that there are home and colonial professorships, and masterships in the secondary schools. Of this there can be no reasonable doubt, because we know that under the present imperfect arrangements quite a number of positions such as these have recently been secured by graduates whose whole University education had been obtained in Edinburgh. The answer to the second question we may give in the national manner, by asking in return if Cambridge of recent years has done anything more than this for her ordinary wranglers. If she has, it is due to nothing else than the lamentable system in vogue which marks out Cambridge as a University, and the so-called Universities of Scotland as Schools.

We often hear it said that, other things being equal, the mathematician who hails from Cambridge has in competitions for professorships and masterships a marked advantage. Very probably this is true; and probably also it has been found to hold when other things were not equal. But in ordinary fairness we must bear in mind that, as matters are at present arranged, when the said other things are not equal, it amounts almost to certainty that the inequality is not due to the inferiority of the Cambridge-trained man. How could it be otherwise? There is a three years' course of mathematical drill at Cambridge not to be matched in any other country of the world. The course may be well planned or ill planned, the drill may be carried out on wise or unwise principles, but there it is, and its like is to be got nowhere else, certainly not in Scotland. So long as this is the case, we must be prepared to see repetitions of the appointments complained against, and must school ourselves to be moderately thankful when, as recently, other Universities get a chance.

This superiority of Cambridge, I have said, is quite independent of the question whether the system pursued there be in its details good or bad. There are many thoughtful mathematicians and other men of science, even among those who have enjoyed a Cambridge training, who are strongly of opinion that it is not by any means wholly good. They deprecate the breakneck pace, and the utilitarian spirit which sacrifices everything to what is known to 'pay' in examinations; and they deplore that almost unique and absolute power it possesses of quenching originality. The perfect man under the Cambridge system is too often, they say, the man who can merely acquire mathematical theorems and methods with rapidity, and reproduce them on demand with like speed. As for the student of high reflective powers or the student of genius, we are asked to believe that unless he is also endowed with great strength of mind and will, he will not come out of the ordeal unscathed. Now, it is mainly students of these two latter types that, for the purposes of research, a University ought to cherish and develop. And my contention - made with all deference - is, that the Scotch graduate who spends three additional years in his own University, following out the course I have indicated, partly and chiefly devoting himself to the study of the higher departments of his subject in continuation of his undergraduate course, partly busying himself with teaching and lecturing, partly engaged in literary work and mathematical investigations, and all this under the eyes of professors earnestly occupied themselves in similar ways - such a graduate, I say, has every chance to turn out a better man for his profession than the graduate of Cambridge under the present regime. We recognise, as has been said, two qualifications for a professorship - skill to instruct and ability for research: it is hard to see where there are now at Cambridge special facilities for obtaining either the one or the other.

When we come to consider next what opportunities the University offers for enabling Professors themselves to engage in research, we are at once met with almost equally startling deficiencies. First and foremost is the extraordinary anachronism of a single professorship for the wide domain of Mathematics. Who that thinks for a moment of the vast additions that have been made to mathematical knowledge, even within the present century, but sees the flagrant inadequacy of this provision? Nothing is more certain than that one man's brain is utterly unable to cope with it; indeed, unless I am much mistaken, the very best professors will be the first to confess that half-a-dozen men will not fully represent it. Mathematics is the last of the sciences to be treated in this way; it is an architectural pile which is being constantly added to, and from which no stone, once laid, is ever removed. But other subjects have been almost quite as much neglected. The so-called Arts Faculties of our Universities remain to-day pretty much as they stood three hundred years ago. What is there to account for the apathy which has left University management to drift on decade after decade as it has done? We had at one time in Scotland an educational system which was excellent for the age in which it existed; it produced good effects on the nation and outsiders were not sparing in their eulogies of it and us: who knows but that the self-complacency which this engendered lulled us into repose and worked our ruin! Whatever the cause may be, certain it is we have slept, and other nations have been thoughtfully on the watch, ready to observe and remedy defects, ready to foresee the coming demand, and to enlarge the boundaries of their institutions. It is scarcely possible to overpraise the admirable singleness of purpose, which some foreign States have shown in working out their schemes of education. Thus, to return to Mathematics, can it be too often brought to notice that a single German University is able to show as many professors of the subject as all the Universities of Scotland put together? Such a statement seems so wild, so like a traveller's tale, to one whose attention is for the first time drawn to it, that we can scarcely make too much of it, hoping that thereby the incredulous may be led to test the truth of it for themselves. Fortunately the test is one that can be easily and agreeably made. Within the space of a summer holiday it is possible for any of us to actually see one of these Universities in the full swing of work. At every one of them there will be found in progress more courses of mathematical lectures than there are professors. The subjects of the courses will be somewhat like the following, which constitute an actual case (1) Differential and Integral Calculus; (2) Definite Integrals; (3) Elliptic Functions; (4) Differential Equations; (5) the Function Theory of Weierstrass; (6) Theory of Equations; (7) Determinants; (8) Curved Surfaces and Curves of Double Curvature: and if the visitor goes back the following session he will not fail to find several courses that are new. How many such courses are within the reach of the students of a Scotch University? nay rather, how many such courses is it possible for any Scotch University to have? It is unwise to try to console ourselves, as is sometimes done, with the statement that the students attending the higher of such courses form but a mere handful. The fact of the existence of the courses would be none the less important, even if this statement were true; but in many cases it is a perversion of the truth - the courses just enumerated were attended by never fewer than fifteen students, and for that on Elliptic Functions there were enrolled considerably over half a hundred. Far be it from me to wish to Germanize our institutions: zeal in this direction is often the zeal that is not of knowledge. So long as there are different nationalities, and especially nationalities with a lengthened past history, so long must their institutions, including their Universities, be cast in different moulds. The Universities of Scotland are in the main admirably suited to the wants of her people: what we have to do is to develop them in the most natural way as time advances and the increase of knowledge demands. The safe wish for us is - to be Scotch in the means we employ, to be more German than the Germans in the magnitude and number of the results we accomplish.

Besides this overburdening of the Scotch Professor, by throwing on his shoulders the full weight of an overgrown subject, there is the aggravation of wasting his energy in teaching the most elementary portions of it. Nothing in connection with our Universities is more astounding to a foreigner than the fact that there are large numbers of students enrolled every year to begin the first proposition of Euclid, and that, of all the mathematical students within the walls, by far the greater portion have confined their studies to elementary Algebra, Geometry and Trigonometry. Fortunately this condition of things is beginning to astound others besides foreigners. Many Scotsmen now confess that it is the one utterly unsightly blot on our system; and that, so long as it continues, there is ground for the reproach that our Universities are Schools. The great attention which has been given to it recently warrants the hope that we are on the eve of something better. The problem presented is doubtless a complicated one, more complicated than many of those suppose who have publicly discussed it. But surely it is not insoluble, when so many clear heads and willing hands might be brought to bear. If the cause of the stigma be the want of secondary schools, then the plain duty of every one, professors as much as any one else, is to help towards the remedying of the defect. If the cause be that the elementary classes in the Universities are a source of income which cannot be dispensed with, the line of action is equally manifest, - every effort should be made, in season and out of season, to have the present incomes of the professors secured to them, indeed, to have the incomes of some of them considerably increased. Is it too Quixotic to suppose that had all mutual recriminations been laid aside, and a National Conference on the Higher Education been held several years ago, - a Conference including professors, secondary schoolmasters, and the many influential laymen who, from their connection with school boards and other public bodies, take an interest in the subject, -a basis of action could have been harmoniously arrived at, which would ere this have made a solution an immediate possibility? The evil is more clamant than ever now, when the far more difficult problem of providing a national system of elementary education has received so satisfactory a solution; and may we not reasonably expect that the intellects and wills which solved the one are capable of solving the other?

There is a third drawback which Scotch professors experience more than professors elsewhere: this is the drain upon their time caused by conducting the purely business affairs of the University. Our transatlantic friends are reported to consider it the acme of genius, enterprise and organising ability, to be able to "run" a newspaper. In these days of competition, local examinations, building schemes, endowment funds, and threatened legislation, we may readily believe that it is no child's play to "run" a University. If a member of Senate be found to have what is called a "good business head," as mathematical professors not unseldom have, he is pretty safe to have put on him not less than his own share of the work that is going.

We may well be disposed, however, to allow this third hindrance to sink quietly into the back-ground, in view of the marked advantage which Arts Professors in Scotland have secured to them, by the extremely equitable division which is made of the civil year, into days of work and days of rest. The policy of having such a lengthened holiday has been more than once called in question. It seems to me that if the argument against it is to be made really effective, the point of view must be the loss of time entailed on the student, rather than the lavishness with which it is gifted to the professor. In a University with a thoroughly adequate staff of mathematical professors, we can quite well imagine a case where a professor, who had greatly distinguished himself in research, might with wisdom be relieved more and more from the drudgery of teaching, and even finally allowed to devote his entire energy to original work. If his zeal in research had prevented him from becoming an adept at teaching, as might not improbably happen, the gain to the University would be two-fold.

In bringing to a close these remarks regarding research in the Universities, I should wish it to be distinctly understood, if it be not already clear from what I have said, that they have been prompted solely by the consciousness that our country is lamentably behind others in this respect, and by the desire that a subject to which I have long been devoted should receive that attention which is its due.

Let us glance now at the SECONDAPY SCHOOLS, and examine what possibilities there are in connection with them for the furtherance of research. In these institutions we have, on a small scale, exactly the same machinery in action, as in the Arts Faculty of a University. The masters are men with a lengthened educational training, much of it in some cases devoted to a particular department of knowledge, and each of them is engaged day by day teaching his own special subject, and that subject alone. Evidently, then, we have here some at least of the conditions which are known to be favourable to the production of literary and scientific work; and we should therefore expect to find that secondary schoolmasters have been accustomed to take their place, - naturally an inferior one, - in helping on the literary and scientific work of a country. Now, in Scotland, it is impossible for us to say that the evidence of this is as strong as it should be. I do not for a moment forget the special cases which go so far to prove the contrary: I cannot, for example, while in Edinburgh, fail to recall the services which such a man as Adam of the High School rendered to research in connection with the Latin language: it is the marked scarcity of names like his that makes the assertion possible. Nor is the explanation of this far to seek. We have never had in Scotland an organised system of secondary schools. There have been individual instances of schools engaged in work quite worthy of the name of secondary instruction, but the instances have been exceedingly few, and as nearly as possible quite independent of each other. No uniform, high standard, necessitating correspondingly high qualifications in the masters, has been sought to be maintained by a controlling power outside the schools. On the contrary, there is evidence in connection with several of them, that the elevation of the standard in any particular subject has been proportioned to the zeal of the teacher of that subject for the time being. The one astonishing fact, indeed, in connection with such schools, is that, the standard in the Universities being so low, it was possible for a single one of them to attain even temporarily to anything like a worthy secondary character at all. The bulk of the University students came for a lengthened period from the parochial schools, and they it was who practically fixed the University standard. This, it seems to me, continues to be the key of the situation. Let the supply of secondary schools in a country be deficient, and the inevitable results are (1) that the University standard becomes suited to the capabilities of the elementary schools, and (2) that the secondary schools actually in existence tend more and more to become "caste" schools, and in this capacity to do elementary work. Doubtless a good deal has been done of recent years by isolated bodies for the improvement of secondary education in Scotland. The fact remains, however, that if we look at the country as a whole, the so-called secondary schools are found to be in great part elementary, the pupils at the lower stages being children of parents who are willing to pay a high fee for elementary instruction, partly, it may be, because of the better style of such instruction which is there given, but also notably in order that their children may associate with others of their own or a higher social class. Who can doubt but that it is this hybrid character of our old Burgh Schools which has hampered school boards in the management of them? If, recognising "caste" or ignoring it, elementary instruction be really sufficiently and properly provided for elsewhere, then the elementary departments of secondary schools are not sources of strength. At present they are sources of strength of a certain kind, but they are at the same time signs of a significant weakness. When the question of the establishment of such a department is discussed, the strong argument in favour is, that it has been tried in such a place and such another place, and has proved a great success. For my part, I should wish that the elements of this success be always carefully stated. Does it not often mean simply and solely a commercial success, - a success exactly analogous to the so-called success which attends the doing of elementary work in the Universities? Painters and sculptors show a praiseworthy attention to the veracities in styling the purely bread-winning efforts of their skill "pot-boilers": the pupils in the lower classes of the Universities and Secondary Schools are "pot-boilers" likewise; and schoolmasters as much as professors require to have their attention drawn to the fact. What is wanted in Scotland is not merely a well-distributed supply of secondary schools of the present type: it is desirable further that the schools be Higher-Grade in reality as well as in name. When this has been attained, there will be found in them a body of men second only to the University Professors, some of them enthusiastic in research by natural inclination, and all of them well-disposed, according to their several abilities, to help on the advancement of knowledge.

If, however, I have been apologetic for professors, because of the inroads made on their time, what shall I say, regarding the leisure of these coming secondary schoolmasters? Shall I plagiarize from an ancient author, and assert roundly, "They will have no leisure." Doubtless, if we may judge from the present, they will at any rate be men with a harder every-day life of professional work and worry, than falls to the lot of University professors. School work and University lecturing, apart from inequality in the length of time consumed by them, are two very different things. As for myself, having had some experience of both, I look back now upon the period of my assistant-professorship as the Golden Age. Of course the reference to gold will be understood as strictly metaphorical, but there were then intervals of time suffused as it were with a holy calm; and time, we know, whether so sanctified or not, is a commodity which gold will not always buy. School life, on the other hand, in these days of compulsory grind, over-examination, and education subordinated to mechanical instruction, one is disposed to look upon as a life of uncommonly little sunshine; and a schoolmaster in charge of a large department with multifarious details, as resembling merely a heated sphere, poised in interstellar space, with energy leaving it at every point. With all drawbacks allowed for, however, we cannot doubt that such a body of men as I have referred to, would be a power for good in the lower lines of mathematical research. They would have kept up, we should hope, their connection with their professors and their University, as is too seldom the case at present, thereby learning occasionally of the work that was going on in their old haunts, hearing of and becoming acquainted with the younger men who were taking to the same paths as themselves, getting news now and again of some particular problem which from their special bent they might take a pleasure in working out, and obtaining at times hints and suggestions as to the lines of least resistance in private investigations of their own. By some such system of action and reaction, vegetation would be more of a rarity than it is among mathematical graduates, and professors would become founders and heads of Schools in a new and better sense.

I should have wished in the next place, gentlemen, to ask your attention for a little to the influence which School and University TEXT-BOOKS exercise in either hindering or promoting the cause of research, pointing out the desirability of making such books less arid, less unsuggestive, and less cunningly designed to secure a "pass," - of arranging the text and exercises in such a way as to develop any germ of originality the reader might possess, - of giving the theorems and methods some human interest by means of historical and bibliographical notes and, above all, of having the higher text-books as nearly as possible accurately representative of the state of knowledge at the date of publication. This, however, as it would have led me to occupy too much of your time, I have been obliged reluctantly to forego.

It was also my intention to have devoted a little space to the consideration of the MATHEMATICAL JOURNALS, in their character as mediums for encouraging and promoting original investigation, - referring to the qualities which had made certain of them markedly influential for good in this direction, - suggesting a number of features which would make an elementary magazine attractive, without interfering with its function as a vehicle for the publication of original matter, - and emphasizing the apparently trivial fact that the receipt of a single such suitable journal at stated intervals would keep alive a young student's mathematical life as few other things could. Under this head there might suitably have come some suggestions I should have liked to make, regarding the possibility of maintaining full sets of the mathematical journals in every University town, and as to what less expensive means might be adopted to obtain a good substitute, where this could not be accomplished. These remarks, also, it has seemed better in the circumstances to omit,

The last and one of the most powerful agencies for the promotion of research, viz., SOCIETIES, I cannot well pass over in this manner, as our own Society was formed with research for one of its objects. It is unnecessary, however, to speak at any length of what societies can accomplish in this way, or of the means they employ in attaining their ends; - these are things of everyday knowledge. One instance only may be taken, because of its special appropriateness to ourselves - the instance of the Mathematical Society of London. Its origin nineteen years ago was quite humble, and not unlike that of our own Society, as the reader of De Morgan's "Life" may learn. Its conception was not due to the great English mathematicians, - great mathematicians do not perhaps feel the want of such aids; but one by one, probably through De Morgan's influence, they too became members and took an active part in its management. At first there were no published Proceedings; by-and-by a specially notable paper or two seemed to demand printing, and printing was begun; and now year by year there appears a goodly volume of close on 300 pages of select matter, the value of which is recognised by all the best mathematicians throughout the world. In looking through these volumes, we see most instructive signs of one at least of the characteristic benefits of a society. We find a paper cropping up in one volume without any apparent immediate progenitor; soon after there appears another paper on the same subject by a new hand; later on a third author takes it up; and so forth. Of this form of "passing on the torch" there is repeated evidence. Another fact, still more striking, is this - that during these nineteen years of the society's life no mathematical journal has gone out of existence, and no older society has published less mathematical work than it did before. The balance indeed is, I believe, on the other side; and we are therefore entitled to say that the series of volumes published by the London Mathematical Society has been a clear gain to science for the period in question. Here, it seems to me, is something for us to emulate. No one, I trust, will say that these things may be done in London, but not here. If we only be true to the self-denying aims which our Society started with, each one "bearing and forbearing" lest the Society should suffer, each one steadfastly and unselfishly working for the advancement of his science, then the success of the London Mathematical Society will assuredly be ours. True, no doubt, that it has all the mathematical giants of the nation for members, and that consequently our work must for a considerable time, perhaps for years, be trivial in comparison. But, gentlemen, we must remember that there is an immensity of work to be done for which giants are wholly unnecessary, and that no work is more useful than preparing the way for the giants of the future. There is room and need for the exertions of every one, down to hewers of wood and drawers of water. I assert with confidence - for I believe that those with more experience than myself will endorse what I say - that there is not a single member of the Edinburgh Mathematical Society but might do useful work in the cause of mathematical research. If only those who have least confidence in their powers were this session to form themselves into small committees with conveners of somewhat more experience, and take up earnestly subjects which had been carefully selected by themselves or chosen for them, next session, I venture to say, would show a marked progress in the Society's work. It has been more than once stated, when the backwardness of English science was being discussed, that of men of the first rank we could point to as great names as any in Europe, and that where the dearth lay was among workers of the third and lower grades. I say for myself, gentlemen, and I hope for others present who have not yet put their hands to the plough, that we are willing to try to fill this gap, and to wipe away the reproach from our country. The little we do will in the fullness of time bear fruit: nothing is more certain than that no part of it can be lost. "A man's little work," says the great Scottish thinker, whom we mathematicians honour none the less highly because he translated Legendre's Geometry, "A man's little work lies not isolated, stranded: a whole busy world environs it: will catch it up: will carry it forward or else backward: always, infallibly, the Thing Done will come to use."

JOC/EFR August 2007

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