MacTutor
Percy MacMahon addresses the British Association in 1901
Percy A MacMahon was President of Section A of the British Association for the Advancement of Science in 1901. The Association met in Glasgow in September and MacMahon addressed Section A - Mathematical and Physical Sciences.
SECTION A. - MATHEMATICAL AND PHYSICAL SCIENCE.
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.
At the beginning of the nineteenth century it was possible for most workers to be well acquainted with nearly all important theories in any division of science; the number of workers was not great, and the results of their labours were for the most part concentrated in treatises and in a few publications especially devoted to science; it was comparatively easy to follow what was being done. At the present time the state of affairs is different. The number of workers is very large; the treatises and periodical scientific journals are very numerous; the ramifications of investigation are so complicated that it is scarcely possible to acquire a competent knowledge of the progress that is being made in more than a few of the subdivisions of any branch of science. Hence the so-called specialist has come into being.
Evident though it be that this is necessarily an age of specialists, it is curious to note that the word 'specialist' is often used as a term of opprobrium, or as a symbol of narrow-mindedness. It has been stated that most specialists run after scientific truth in intellectual blinkers; that they wilfully restrain themselves from observing the work of others who may be even in the immediate neighbourhood; that even when the line of pursuit intersects obviously other lines, such intersection is passed by without remark; that no attention is paid to the existence or the construction of connecting lines; that the necessity for collaboration is overlooked; that the general advance of the body of scientific truth is treated as of no concern; that absolute independence of aim is the thing most to be desired. I propose to inquire into the possibility of such an individual existing as a scientific man.
I take as a provisional definition of a specialist in science one who devotes a very large proportion of his energies to original research in a particular subdivision of his subject. It will be sufficient to consider the subjects that come under the purview of Section A, though it will be obvious that a similar train of reasoning would have equal validity in connection with the subjects included in any of the other sections. I take the word 'specialist' to denote a man who makes original discoveries in some branch or science, and I deny that any other man has the right, in the modern meaning of the word, to be called by others, or to call himself, a specialist. I would not wish to be understood to imply a belief that a truly scientific man is necessarily a specialist; I do believe that a scientific man of high type is almost invariably an original discoverer in one or more special branches of science; but I can conceive that a man may study the mutual relations of different sciences and of different branches of the same science and may throw such an amount of light upon the underlying principles as to be in the highest degree scientific. I will now advance the proposition that, with this exception, all scientific workers are specialists; it is merely a question of degree. An extreme specialist is that man who makes discoveries in only one branch, perhaps a very narrow branch, of his subject. I shall consider that in defending him I am à fortiori defending the man who is a specialist, but not of this extreme character.
A subject of study may acquire the reputation of being narrow either because it has for some reason or other not attracted workers, and is in reality virgin soil only awaiting the arrival of a husbandman with the necessary skill; or because it is an extremely difficult subject which has resisted previous attempts to elucidate it. In the latter case, it is not likely that a scientific man will obstinately persist in trying to force an entrance through a bare blank wall. Either from weariness in striving, or from the exercise of his judgment, he will turn to some other subdivision which appears to give greater promise of success. When the subject is narrow merely because it has been overlooked, the specialist has a grand opportunity for widening and freeing it from the reproach of being narrow; when it is narrow from its inherent difficulty he has the opportunity of exerting his full strength to pierce the barriers which close the way to discoveries. In either case the specialist, before he can determine the particular subject which is to engage his thoughts, must have a fairly wide knowledge of the whole of his subject. If be does not possess this be will most likely make a bad choice of particular subjects, or, having made a wise selection, will lack an essential part of the mental equipment necessary for a successful investigation. Again, though the subject may be a narrow one, it by no means follows that the appropriate or possible methods of research are prescribed within narrow limits. I will instance the Theory of Numbers which, in comparatively recent times, was a subject of small extent and of restricted application to other branches of science. The problems that presented themselves naturally, or were brought into prominence by the imaginations of great intellects, were fraught with difficulty. There seemed to be an absence, partial or complete, of the law and order that investigators had been accustomed to find in the wide realm of continuous quantity. The country as explored was found to be full of pitfalls for the unwary. Many a lesson concerning the danger of hasty generalisation had to be learnt and taken to heart. Many a false step had to be retraced. Many a road which a first reconnaissance had shown to be straight for a short distance, was found on further exploration, to suddenly change its direction and to break up into a number of paths which wandered in a fitful manner in country of increasing natural difficulty. There were few vanishing points in the perspective. Few, also, and insignificant were the peaks from which a general view could be gathered of any considerable portion of the country. The surveying instruments were inadequate to cope with the physical characters of the land. The province of the Theory of Numbers was forbidding. Many a man returned empty-handed and baffled from the pursuit, or else was drawn into the vortex of a kind of Maelström and had his heart crushed out of him. But early in the last century the dawn of a brighter day was breaking. A combination of great intellects Legendre, Gauss, Eisenstein, Stephen Smith, etc. - succeeded in adapting some of the existing instruments of research in continuous quantity to effective use in discontinuous quantity. These adaptations are of so difficult and ingenious a nature that they are to-day, at the commencement of a new century, the wonder and, I may add, the delight of beholders. True it is that the beholders are few. To attain to the point of vantage is an arduous task demanding alike devotion and courage. I am reminded, to take a geographical analogy, of the Hamilton Falls, near Hamilton Inlet, in Labrador. I have been informed that to obtain a view of this wonderful natural feature demands so much time and intrepidity, and necessitates so many collateral arrangements, that a few years ago only nine white men had feasted their eyes on falls which are finer than those of Niagara. The labours of the mathematicians named have resulted in the formation of a large body of doctrine in the Theory of Numbers. Much that, to the superficial observer, appears to lie on the threshold of the subject is found to be deeply set in it and to be only capable of attack after problems at first sight much more complicated have been solved. The mirage that distorted the scenery and obscured the perspective has been to some extent dissipated; certain vanishing points have been ascertained; certain elevated spots giving extensive views have been either found or constructed. The point I wish to urge is, that these specialists in the Theory of Numbers were successful for the reason that they were not specialists at all in any narrow meaning of the word. Success was only possible because of the wide learning of the investigator; because of his accurate knowledge of the instruments that had been made effective in other branches; and because he had grasped the underlying principles which caused those instruments to be effective in particular cases. I am confident that many a worker who, from the supposed extremely special character of his researches has been the mark of sneer and of sarcasm, would be found to have devoted the larger portion of his time to the study of methods which had been available in other branches, perhaps remote from the one which was particularly attracting his attention. He would be found to have realised that analogy is often the finger-post that points the way to useful advance; that his mind had been trained, and his work assisted, by studying exhaustively the successes and failures of his fellow-workers. But it is not only existing methods that may be available in a special research.
Furthermore, a special study frequently creates new methods which may be subsequently found applicable to other branches. Of this the Theory of Numbers furnishes several beautiful illustrations. Generally, the method is more important than the immediate result. Though the result is the offspring of the method, the method is the offspring of the search after the result. The Law of Quadratic Reciprocity, a corner-stone of the edifice, stands out not only for the influence it has exerted in many branches, but also for the number of new methods to which it has given birth, which are now a portion of the stock-in-trade of a mathematician. Euler, Legendre, Gauss, Eisenstein, Jacobi, Kronecker, Poincaré, and Klein are great names that will be for ever associated with it. Who can forget the work of H J S Smith on homogeneous forms and on the five-square theorem, work which gave rise to processes that have proved invaluable over a wide field, and which supplied many connecting links between departments which were previously in more or less complete isolation?
In this connection I will further mention two branches with which I have a more special acquaintance - the theory of invariants, and the combinatorial analysis. The theory of invariants was evolved by the combined efforts of Boole, Cayley, Sylvester, and Salmon, and has progressed during the last sixty years with the co-operation, amongst others, of Aronhold, Clebsch, Gordan, Brioschi, Lie, Klein, Poincaré, Forsyth, Hilbert, Elliott, and Young. It involves a principle which is of wide significance in all the subject-matters of inorganic science or organic science, and of mental, moral and political philosophy. In any subject of inquiry there are certain entities, the mutual relations of which under various conditions it is desirable to ascertain. A certain combination of these entities may be found to have an unalterable value when the entities are submitted to certain processes or are made the subjects of certain operations. The theory of invariants in its widest scientific meaning determines these combinations, elucidates their properties, and expresses results when possible in terms of them. Many of the general principles of political science and economics can be expressed by means of invariantive relations connecting the factors which enter as entities into the special problems. The great principle of chemical science which asserts that when elementary or compound bodies combine with one another the total weight of the materials is unchanged, is another case in point. Again, in physics, a given mass of gas under the operation of varying pressure and temperature has the well-known invariant, pressure multiplied by volume and divided by absolute temperature. Examples might be multiplied. In mathematics the entities under examination may be arithmetic, algebraic, or geometric; the processes to which they are subjected may be any of those which are met with in mathematical work. It is the principle which is so valuable. It is the idea of invariance that pervades to-day all branches of mathematics. It is found that in investigations the invariantive fractions are those which persist in presenting themselves, even when the processes involved are not such as to ensure the invariance of those functions. Guided by analogy may we not anticipate similar phenomena in other fields of work?
The combinatorial analysis may be described as occupying an extensive region between the algebras of discontinuous and continuous quantity. It is to a certain extent a science of enumeration, of measurement by means of integers, as opposed to measurement of quantities which vary by infinitesimal increments. It is also concerned with arrangements in which differences of quality and relative position in one, two, or three dimensions, are factors. Its chief problem is the formation of connecting roads between the sciences of discontinuous and continuous quantity. To enable, on the one hand, the treatment of quantities which vary per saltum, either in magnitude or position, by the methods of the science of continuously varying quantity and position, and on the other hand to reduce problems of continuity to the resources available for the management of discontinuity. These two roads of research should be regarded as penetrating deeply into the domains which they connect.
In the early days of the revival of mathematical learning in Europe the subject of 'combinations' cannot be said to have rested upon a scientific basis. It was brought forward in the shape of a number of isolated questions of arrangement, which were solved by mere counting. Their solutions did not further the general progress, but were merely valuable in connection with the special problems. Life and form, however, were infused when it was recognised by De Moivre, Bernoulli, and others that it was possible to create a science of probability on the basis of enumeration and arrangement. Jacob Bernoulli, in his Ars Conjectandi, 1713, established the fundamental principles of the Calculus of Probabilities. A systematic advance in certain questions which depend upon the partitions of numbers was only possible when Euler showed that the identity reduced arithmetical addition to algebraical multiplication and vice versa. Starting with this notion, Euler developed a theory of generating functions on the expansion of which depended the formal solutions of many problems. The subsequent work of Cayley and Sylvester rested on the same idea, and gave rise to many improvements. The combinations under enumeration had all to do with what may be termed arrangements on a line subject to certain laws. The results were important algebraically as throwing light on the theory of Algebraic series, but another large class of problems remained untouched, and was considered as being both outside the scope and beyond the power of the method. I propose to give some account of these problems, and to add a short history of the way in which a method of solution has been reached. It will be gathered from remarks made above that I regard any department of scientific work, which seems to be narrow or isolated, as a proper subject for research. I do not believe in any branch of science, or subject of scientific work, being destitute of connection with other branches. If it appears to be so, it is especially marked out for investigation by the very unity of science. There is no necessarily pathless desert separating different regions. Now a department of pure mathematics which appeared to be somewhat in this forlorn condition a few years ago, was that which included problems of the nature of the magic square of the ancients. Conceive a rectangular lattice or generalised chess board (cf. 'Gitter,' Klein), whose compartments are situations for given numbers or quantities, so that there is a rectangular array of certain entities. The general problem is the enumeration of the arrays when both the rows and the columns of the lattice satisfy certain conditions. With the simplest of such problems certain progress had undoubtedly been made. The article on Magic Squares in the Encyclopaedia Britannica, and others on the same subject in various scientific publications, are examples of such progress, but the position of isolation was not sensibly ameliorated. Again the well-known 'problème des rencontres' is an instance in point. Here the problem is to place a number of different entities in an assigned order in a line and beneath them the same entities in a different order subject to the condition that the entities in the same vertical line are to be different. This easy question has been solved by generating functions, finite differences, and in many other ways. In fact when the number of rows is restricted to two, the difficulties inherent in the problem when more than two rows are in question do not present themselves. The problem of the Latin Square is concerned with a square of order and different quantities which have to be placed one in each of the compartments in such wise that each row and each column contains each of the quantities. The enumeration of such arrangements was studied by mathematicians from Euler to Cayley without any real progress being made. In reply to the remark 'Cui bono?' I should say that such arrangements have presented themselves for investigation in other branches of mathematics. Symbolical algebras, and in particular the theory of discontinuous groups of operations, have their laws defined by what Cayley has termed a multiplication table. Such multiplication tables are necessarily Latin Squares, though it is not conversely true that every Latin Square corresponds to a multiplication table. One of the most important questions awaiting solution in connection with the theory of finite discontinuous groups is the enumeration of the types of groups of given order, or of Latin Squares which satisfy additional conditions. It thus comes about that the subject of Latin Squares is important in mathematics, and some new method of dealing with them seems imperative.
A fundamental idea was that it might be possible to find some mathematical operation of which a particular Latin Square might be the diagrammatic representative. If, then, a one-to-one correspondence could be established between such mathematical operations and the Latin Squares, the enumeration might conceivably follow. Bearing this notion in mind, consider the differentiation of with regard to . Noticing that the result is ( an integer), let us inquire whether we can break up the operation of differentiation into elementary portions, each of which will contribute a unit to the resulting coefficient . If we write down as the product of n letters, viz., , it is obvious that if we substitute unity in place of a single in all possible ways, and add together the results, we shall obtain . We have, therefore, different elementary operations, each of which consists in substituting unity for . We may denote these diagrammatically by
and from this point of view is a combinatorial symbol, and denotes by the coefficient the number of ways of selecting one out of different things.
Similarly, the higher differentiations give rise to diagrams of two or more rows, the numbers of which are given by the coefficients which result from such differentiations. Following up this clue much progress has been made. For a particular problem success depends upon the design, on the one hand, of a function, on the other hand, of an operation such that diagrams make their appearance which have a one-to-one correspondence with the entities whose enumeration is sought. For a general investigation, however, it is more scientific to start by designing functions and operations, and then to ascertain the problems of which the solution is furnished. The difficulties connected with the Latin Square and with other more general questions have in this way been completely overcome.
The second new method in analysis that I desire to bring before the Section had its origin in the theory of partition. Diophantus was accustomed to consider algebraical questions in which the symbols of quantity were subject to certain conditions, such, for instance, that they must denote positive numbers or integer numbers. A usual condition with him was that the quantities must denote positive integers. All such problems and particularly those last specified are qualified by the adjective Diophantine. The partition of numbers is then on all fours with the Diophantine equation
a further condition being that one solution only is given by a group of numbers satisfying the equation; that in fact permutations amongst the quantities are not to be taken into account. This further condition is brought in analytically by adding the Diophantine inequalities
in number. The importation of this idea leads to valuable results in the theory of the subject which suggested it. A generating function can be formed which involves in its construction the Diophantine equation and inequalities, and leads after treatment to a representative, as well as enumerative, solution of the problem. It enables further the establishment of a group of fundamental parts of the partitions from which all possible partitions of numbers can be formed by addition with repetition. In the case of simple unrestricted partition it gives directly the composition by rows of units which is in fact carried out by the Ferrers-Sylvester graphical representation, and led in the hands of the latter to important results connection with algebraical series which present themselves in elliptic functions and in other departments of mathematics. Other branches of analysis and geometry supply instances of the value of extreme specialisation.
What we require is not the disparagement of the specialist, but the stamping out of narrow-mindedness and of ignorance of the nature of the scientific spirit and of the life-work of those who devote their lives to scientific research. The specialist who wishes to accomplish work of the highest excellence must be learned in the resources of science and have constantly in mind its unity and its grandeur.
Last Updated April 2007