To read the second part of MacMahon's lecture, follow the link: British Association 1901, Part 2
To read the second part of MacMahon's lecture, follow the link: British Association 1901, Part 2
PRESIDENT OF THE SECTION. - Major P A MACMAHON, D.Sc., F.R.S.
THURSDAY, 12 SEPTEMBER 1901.
Percy A MacMahon, the President, delivered the following Address:
During the seventy meetings of the Association a pure mathematician has been president of Section A on ten or a dozen occasions. A theme taken by many has been a defence of the study of pure mathematics. I take Cayley's view expressed before the whole Association at Southport in 1883 that no defence is necessary, but were it otherwise I feel that nothing need be added to the eloquent words, of Sylvester in 1869 and of Forsyth in 1897. I intend therefore to make some remarks on several matters which may be interesting to the Section even at the risk of being considered unduly desultory.
Before commencing I must remark that during the twelve months that have elapsed since the Bradford Meeting we have lost several great men whose lives were devoted to the subjects of this Section. Hermite, the veteran mathematician of France, has left behind him a splendid record of purely scientific work. His name will be always connected with the Herculean achievement of solving the general quintic equation by means of elliptic modular functions. Other work, if less striking, is equally of the highest order, and his treatise 'Cours d'Analyse' is a model of style. Of FitzGerald of Dublin it is not easy to speak in this room without emotion. For many years he was the life and soul of this Section. His enthusiasm in regard to all branches of molecular physics, the force and profundity of his speech, the vigour of his advocacy of particular theories, the active thinking which enabled him to formulate desiderata, his warm interest in the work of others, and the unselfish aid he was so willing to give, are fresh in our remembrance. Rowland was in the forefront of the ranks of physicists. His death at a comparatively early age terminates the important series of discoveries which were proclaimed from his laboratory in the Johns Hopkins University at Baltimore. In Viriamu Jones we have lost an assiduous worker at physics whose valuable contributions to knowledge indicated his power to do much more for science. In Tait, Scotland possessed a powerful and original investigator. The extent and variety of his papers are alike remarkable, and in his collected works there exists an imperishable monument, to his fame.
It is interesting in this the first year of the new century, to take a rapid glance at the position that mathematicians of this country held amongst mathematicians a hundred years ago. During the greater part of the eighteenth century the study of mathematics in England, Scotland and Ireland had been at a very low ebb. Whereas in 1801 on the Continent there were the leaders Lagrange, Laplace and Legendre, and of rising men, Fourier, Ampère, Poisson and Gauss, we could only claim Thomas Young and Ivory as men who were doing notable work in research. Amongst schoolboys of various ages we note Fresnel, Bessel, Cauchy, Chasles, Lamé, Möbius, von Staudt and Steiner on the Continent, and Babbage, Peacock, John Herschel, Henry ParrHamilton and George Green in this country. It was not indeed till about 1845 or a little later that we could point to the great names of William Rowan Hamilton, MacCullagh, Adams, Boole, Salmon, Stokes, Sylvester, Cayley, William Thomson, H J S Smith and Clerk Maxwell as adequate representatives of mathematical science. It is worthy of note that this date, 1845, marks also the year of the dissolution of a very interesting society, the Mathematical Society of Spitalfields; and I would like to pause a moment and, if I may say so, rescue it from the oblivion which seems to threaten it. In 1801 it was already a venerable institution, having been founded by Joseph Middleton, a writer of mathematical text-books, in 1717. [Its first place of meeting was the Monmouth's Head, Monmouth Street, Spitalfields. This street has long disappeared. From a map of London of 1746 it appears to have run parallel to the present Brick Lane and to have corresponded to the present Wilks Street.] The members of the Society at the beginning were for the most part silk-weavers of French extraction; it was little more than a working man's club at which questions of mathematics and natural philosophy were discussed every Saturday evening. The number of members was limited to the 'square of seven,' but later it was increased to the 'square of eight,' and later still to the 'square of nine.' In 1725 the place of meeting was changed from the Monmouth's Head to the White Horse in Wheeler Street, and in 1735 to the Ben Jonson's Head in Pelham Street. The subscription was six-and-sixpence a quarter, or sixpence a week, and entrance was gained by production of a metal ticket which had the proposition of Pythagoras engraved on one side and a sighted quadrant with level on the other. The funds, largely augmented by an elaborate system of fines, were chiefly used for the purchase of books and physical apparatus. A president, treasurer, inspector of instruments and secretary were appointed annually, and there were, besides, four stewards, six auditors, and six trustees. By the constitution of the Society it was the duty of every member, if he were asked any mathematical or philosophical question by another member, to instruct him to the best of his ability. It was the custom for each member in rotation to lecture or perform experiments at each evening meeting. There was a fine of half-a-crown for introducing controverted points of divinity or politics. The members dined together twice annually, viz., on the second Friday in January in London in commemoration of the birth of Sir Isaac Newton (this feast frequently took place at the Black Swan, Brown's Lane, Spitalfields), and on the Second Friday in July 'at a convenient distance in the country in commemoration of the birth of the founder.' The second dinner frequently fell through because the members could not agree as to the locality. It was found necessary to introduce a rule fining members sixpence for letting off fireworks in the place of meeting. Every member present was entitled to a pint of beer at the common expense, and, further, every five members were entitled to call for a quart for consumption at the meeting. Such were some of the quaint regulations in force when, about the year 1750, the Society moved to larger apartments in Crispin Street, where it remained without interruption till 1843. It appears from the old minute books that about the year 1750 the Society absorbed a small mathematical society which used to meet at the Black Swan, Brown's Lane, above mentioned, and that in 1783 an ancient historical society was also incorporated with it. By the year 1800 the class of the members had become improved, and we find some well-known names, such as Dolland, Simpson, Saunderson, Crossley, Paroissen and Gompertz. At this time lectures were given in all branches of science by the members in the Society's rooms, which on these occasions were open to the public on payment of one shilling. The arrangements for the session 1822-23 included lectures in mechanics, hydrostatics and hydraulics, pneumatics, optics, astronomy, chemistry, electricity, galvanism, magnetism and botany, illustrated by experiments. On account of these lectures the Society had to fight an action-at-law, and although the case was won, its slender resources were crippled for many years. In 1827 Benjamin Gompertz, F.R.S., succeeded to the presidency on the death of the Rev George Paroissen. From the year 1830 onwards the membership gradually declined and the financial outlook became serious. In 1843 there was a crisis; the Society left Crispin Street for cheaper rooms at 9 Devonshire Street, Bishopsgate Street, and finally, in 1845, after a futile negotiation with the London Institution, it was taken over by the Royal Astronomical Society, which had been founded in 1821. The library and documents were accepted and the few surviving members were made life members of the Astronomical Society without payment. So perished this curious old institution; it had amassed a really valuable library, containing books on all branches of science. The Astronomical Society has retained the greater part, but some have found their way to the libraries of the Chemical and other societies. An inspection of the documents establishes that it was mainly a society devoted to physics, chemistry and natural history. It had an extensive museum of curiosities and specimens of natural history, presented by individual members, which seems to have disappeared when the rooms in Crispin Street were vacated. It seems a pity that more effort was not made to keep the old institution alive, The fact is that at that date the Royal Society had no sympathy with special societies and did all in its power to discourage them. The Astronomical Society was only formed in 1821 in the teeth of the opposition of the Royal Society.
Reverting now to the date 1845 it may be said that from this period to 1866 much good work emanated from this country, but no Mathematical Society existed in London. At the latter date the present Society was formed, with De Morgan as its first President. Gompertz was an original member, and the only person who belonged to both the old and new societies. The thirty-three volumes of proceedings that have appeared give a fair indication of the nature of the mathematical work that has issued from the pens of our countrymen. All will admit that it is the duty of anyone engaged in a particular line of research to keep himself abreast of discoveries, inventions, methods, and ideas, which are being brought forward in that line in his own and other countries. In pure science this is easier of accomplishment by the individual worker than in the case of applied science. In pure mathematics the stately edifice of the Theory of Functions has, during the latter part of the century which has expired, been slowly rising from its foundations on the continent of Europe. It had reached a considerable height and presented an imposing appearance before it attracted more than superficial notice in this country and in America. It is satisfactory to note that during recent years much of the leeway has been made up. English speaking mathematicians have introduced the first notions into elementary textbooks; they have written advanced treatises on the whole subject; they have encouraged the younger men to attend courses of lectures in foreign universities; so that to-day the best students in our universities can attend courses at home given by competent persons, and have the opportunity of acquiring adequate knowledge, and of themselves contributing to the general advance. The Theory of Functions, being concerned with the functions that satisfy differential equations, has attracted particularly the attention of those whose bent seemed to be towards applied mathematics and mathematical physics, and there is no doubt, in analogy with the work of Poincaré in celestial dynamics, those sciences will ultimately derive great benefit from the new study. If, on the other hand, one were asked to specify a department of pure mathematics which has been treated somewhat coldly in this country during the last quarter of the last century, one could point to geometry in general, and to pure geometry, descriptive geometry, and the theory of surfaces in particular. This may doubtless be explained by the circumstance that, at the present time, the theory of differential equations and the problems that present themselves in their discussion are of such commanding importance from the point of view of the general advance of mathematical science that those subjects naturally prove to be most attractive.
As regards organisation and co-operation in mathematics, Germany, I believe, stands first. The custom of offering prizes for the solutions of definite problems which are necessary to the general advance obtains more in Germany and in France than here, where, I believe, the Adams Prize stands alone. The idea has an indirect value in pointing out some of the more pressing desiderata to young and enthusiastic students, and a direct importance in frequently, as it proves, producing remarkable dissertations on the proposed questions. The field is so vast that any comprehensive scheme of co-operation is scarcely possible, though much more might be done with advantage.
If we turn our eyes to the world of astronomy we find there a grand scheme of co-operation which other departments may indeed envy. The gravitation formula has been recognised from the time of Newton as ruling the dynamics of the heavens, and the exact agreement of the facts derived from observation with the simple theory has established astronomy as the most exact of all the departments of applied science. Men who devote themselves to science are actuated either by a pure love of truth or because they desire to apply natural knowledge to the benefit of mankind. Astronomers belong, as a rule, to the first category, which, it must be admitted, is the more purely scientific. We not only find international co-operation in systematically mapping the universe of stars and keeping all portions of the universe under constant observation, but also when a particular object in the heavens presents itself under circumstances of peculiar interest or importance, the observatories of the world combine to ascertain the facts in a manner which is truly remarkable. As an illustration, I will instance the tiny planet Eros discovered a few years ago by De Witt. Recently the planet was in opposition and more favourably situated for observation than it will be again for thirty years. It was determined, at a conference held in Paris in July 1900, that combined work should be undertaken by no fewer than fifty observatories in all parts of the world. Beyond the fixing of the elements of the mean motion and of the perturbations of orbit due to the major planets, the principal object in view is the more accurate determination of solar parallax. To my mind this concert of the world, this cosmopolitan association of fine intellects, fine instruments, and the best known methods, is a deeply impressive spectacle and a grand example of an ideal scientific spirit. Other sciences are not so favourably circumstanced as is astronomy for work of a similar kind undertaken in a similar spirit. If in comparison they appear to be in a chaotic state, the reason in part must be sought for in conditions inherent to their study, which make combined work more difficult, and the results of such combined work as there is, less striking to spectators. Still, the illustration I have given is a useful object-lesson to all men of science, and may encourage those who have the ability and the opportunity to make strenuous efforts to further progress by bringing the work of many to a single focus.
In pure science we look for a free interchange of ideas, but in applied physics the case is different, owing to the fact that the commercial spirit largely enters into them. In a recent address, Professor Perry has stated that the standard of knowledge in electrical engineering in this country is not as high as it is elsewhere, and all men of science and many men in the street know him to be right. This is a serious state of affairs, to which the members of this Section cannot be in any sense indifferent. We cannot urge that it is a matter with which another Section of the Association is concerned to a larger degree. It is our duty to take an active, and not merely passive attitude towards this serious blot on the page of applied science in England. For this many reasons might be given, but it is sufficient to instance one, and to state that neglect of electrical engineering has a baneful effect upon research in pure science in this country. It hinders investigations in pure physics by veiling from observation new phenomena which arise naturally, and by putting out of our reach means of experimenting with new combinations on a large scale. Professor Perry has assigned several reasons for the present impasse, viz., a want of knowledge of mathematics on the part of the rising generation of engineers; the bad teaching of mathematics; the antiquated methods of education generally; and want of recognition of the fact that engineering is not on stereotyped lines, but, in its electrical aspect, is advancing at a prodigious rate; municipal procrastination, and so on. He confesses, moreover, that he does not see his way out of the difficulty, and is evidently in a condition of gloomy apprehension.
It is, I think, undoubted that science has been neglected in this country, and that we are reaping as we have sowed. The importance of science teaching in secondary schools has been overlooked. Those concerned in our industries have not seen the advantage of treating their workshops and manufactories as laboratories of research. The Government has given too meagre an endowment to scientific institutions, and has failed to adequately encourage scientific men and to attract a satisfactory quota of the best intellects of the country to the study of science. Moreover, private benefactors have not been so numerous as in some other countries in respect of those departments of scientific work which are either non-utilitarian or not immediately and obviously so. We have been lacking alike in science organisation and in effective co-operation in work.
It has been attempted to overcome defects in training for scientific pursuits by the construction of royal roads to scientific knowledge. Engineering students have been urged to forego the study of Euclid, and, as a substitute, to practise drawing triangles and squares; it has been pointed out to them that mathematical study has but one object, viz., the practical carrying out of mathematical operations; that a collection of mathematical rules of thumb is what they should aim at; that a knowledge of the meaning of processes may be left out of account so long as a sufficient grasp of the application of the resulting rules is acquired. In particular, it has been stated that the study of the fundamental principles of the infinitesimal calculus may profitably be deferred indefinitely so long as the student is able to differentiate and integrate a few of the simplest functions that are met with in pure and applied physics. The advocates of these views are, to my mind, urging a process of 'cramming' for the work of life which compares unfavourably with that adopted by the so-called 'crammers' for examinations; the latter I believe to be, as a rule, much maligned individuals, who succeed by good organisation, hard work, and personal influence, where the majority of public and private schools fail; the examinations for which their students compete encourage them to teach their pupils to think, and not to rely principally upon remembering rules. The best objects of education, I believe, are the habits of thought and observation, the teaching of how to think, and the cultivation of the memory; and examiners of experience are able to a considerable extent to influence the teaching in these respects; they show the teachers the direction in which they should look for success. The result has been that the 'crammer' for examinations, if he ever existed, has disappeared. But what can be said for the principle of cramming for the work of one's life? Here an examination would be no check, for examiners imbued with the same notion would be a necessary part of the system; the awakening of the student would come, perhaps slowly, but none the less inevitably; he might exist for a while on his formulae and his methods, but with the march of events, resulting in new ideas, new apparatus, new designs, new inventions, new materials requiring the utmost development of the powers of the mind, he will certainly find himself hopelessly at sea and in constant danger of discovering that he is not alone in thinking himself an impostor. And an impostor he will be if he does not by his own assiduity cancel the pernicious effects of the system upon which he has been educated. I do not, I repeat, believe in royal roads, though I appreciate the advantage of easy coaches in kindred sciences. In the science to which a man expects to devote his life, the progress of which he hopes to further, and in which he looks for his life's success, there is no royal road. The neglect of science is not to be remedied by any method so repugnant to the scientific spirit; we must take the greater, knowing that it includes the less, not the less, hoping that in some happy-go-lucky way the greater will follow.
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