## 1912 International Congress of Mathematicians - Cambridge, England

The International Congress of Mathematicians was held in Cambridge, England from 22 August to 28 August 1912. There were 574 full members, 134 family members, 708 total. We give below a version of:

Before presenting the material, we give a short Preface.

Before presenting the material, we give a short Preface.

**Preface by EFR and JJOC.**

At the Cambridge, England, Congress as at the previous Congress in Rome, there was a strong emphasis on applications. Four of the eight plenary lectures are on applications of mathematics. The Regulations for the Congress were published in English, French, German and Italian. Indeed, of the four pure mathematics lectures, one was in each of these four languages. The report of the International Commission on the Teaching of Mathematics showed remarkable achievements despite the difficult political situation which would lead to World War I two years after this Congress. The 1916 Congress was awarded to Stockholm and invitations for 1920 in Budapest and 1924 in Athens were made but, in line with the regulations, only the 1916 Congress was definite. The outbreak of World War I in 1914 meant these all fell by the wayside.

**1. Regulations for the Fifth International Congress of Mathematicians.**

I. At the first General Meeting the Chair shall be taken by the President of the Organising Committee. The Meeting shall proceed to elect the following Officers of the Congress:

A President,

Vice-Presidents,

General Secretaries.

II. The President of the Congress or one of the Vice-Presidents shall preside at each of the succeeding General Meetings.

III. One of the Introducers nominated by the Organising Committee shall preside at the first Meeting of each Section. At such Meeting the Section shall appoint a Secretary and one or more Assistant Secretaries. The Secretaries shall remain in office for the whole time of the Congress. At each Meeting the members present shall elect the President for the next Meeting.

IV. The Organising Committee shall settle the order in which the communications to each Section shall be read. This order may however be modified by a vote of the Section concerned.

V. The reading of a Communication shall not occupy more than twenty minutes. During a discussion a speaker shall not be allowed more than ten minutes, nor shall he speak more than once on the same subject without special permission from the President of the Section.

VI. The speakers are requested to furnish the Secretary of the Section with a brief resume of their remarks immediately after the conclusion of the discussion. The Sectional Secretary, at the end of each Meeting, shall draw up and send to the General Secretary the titles of the papers read for publication in the Journal of the following day. A complete report containing the abstracts of the Communications and of the subsequent discussions shall be drawn up by the Secretary of the Section before the end of the Congress.

VII. The Lectures and Communications read at the Congress shall be collected in the Volume of Proceedings. Authors should deliver the texts of their Lectures and Communications to the General Secretary of the Congress not later than the end of the Congress. Those Lectures or Communications which are written in French, German, or Italian should be type-written (except formulae).

A President,

Vice-Presidents,

General Secretaries.

II. The President of the Congress or one of the Vice-Presidents shall preside at each of the succeeding General Meetings.

III. One of the Introducers nominated by the Organising Committee shall preside at the first Meeting of each Section. At such Meeting the Section shall appoint a Secretary and one or more Assistant Secretaries. The Secretaries shall remain in office for the whole time of the Congress. At each Meeting the members present shall elect the President for the next Meeting.

IV. The Organising Committee shall settle the order in which the communications to each Section shall be read. This order may however be modified by a vote of the Section concerned.

V. The reading of a Communication shall not occupy more than twenty minutes. During a discussion a speaker shall not be allowed more than ten minutes, nor shall he speak more than once on the same subject without special permission from the President of the Section.

VI. The speakers are requested to furnish the Secretary of the Section with a brief resume of their remarks immediately after the conclusion of the discussion. The Sectional Secretary, at the end of each Meeting, shall draw up and send to the General Secretary the titles of the papers read for publication in the Journal of the following day. A complete report containing the abstracts of the Communications and of the subsequent discussions shall be drawn up by the Secretary of the Section before the end of the Congress.

VII. The Lectures and Communications read at the Congress shall be collected in the Volume of Proceedings. Authors should deliver the texts of their Lectures and Communications to the General Secretary of the Congress not later than the end of the Congress. Those Lectures or Communications which are written in French, German, or Italian should be type-written (except formulae).

**2. Proceedings of the Congress.**

*Wednesday,*21

*August*.

At 21.30 the Members of the Congress were received by Sir George Howard Darwin, President of the Cambridge Philosophical Society, and were presented to Mr R F Scott, Vice-Chancellor of the University, at a conversazione held in the Combination Room and Hall of St John's College.

*Thursday,*22

*August*.

The opening meeting of the Congress was held at 10.00.

Sir G H Darwin, President of the Cambridge Philosophical Society, spoke as follows:

Four years ago at our Conference at Rome the Cambridge Philosophical Society did itself the honour of inviting the International Congress of Mathematicians to hold its next meeting at Cambridge. And now I, as President of the Society, have the pleasure of making you welcome here. I shall leave it to the Vice-Chancellor, who will speak after me, to express the feeling of the University as a whole on this occasion, and I shall confine myself to my proper duty as the representative of our Scientific Society.

The Science of Mathematics is now so wide and is already so much specialised that it may be doubted whether there exists today any man fully competent to understand mathematical research in all its many diverse branches. I, at least, feel how profoundly ill-equipped I am to represent our Society as regards all that vast field of knowledge which we classify as pure mathematics. I must tell you frankly that when I gaze on some of the papers written by men in this room I feel myself much in the same position as if they were written in Sanskrit.

But if there is any place in the world in which so one-sided a President of the body which has the honour to bid you welcome is not wholly out of place it is perhaps Cambridge. It is true that there have been in the past at Cambridge great pure mathematicians such as Cayley and Sylvester, but we surely may claim without undue boasting that our University has played a conspicuous part in the advance of applied mathematics. Newton was as a glory to all mankind, yet we Cambridge men are proud that fate ordained that he should have been Lucasian Professor here. But as regards the part played by Cambridge I refer rather to the men of the last hundred years, such as Airy, Adams, Maxwell, Stokes, Kelvin, and other lesser lights, who have marked out the lines of research in applied mathematics as studied in this University. Then too there are others such as our Chancellor, Lord Rayleigh, who are happily still with us.

Up to a few weeks ago there was one man who alone of all mathematicians might have occupied the place which I hold without misgivings as to his fitness; I mean Henri Poincaré. It was at Rome just four years ago that the first dark shadow fell on us of that illness which has now terminated so fatally. You all remember the dismay which fell on us when the word passed from man to man "Poincaré is ill." We had hoped that we might again have heard from his mouth some such luminous address as that which he gave at Rome; but it was not to be, and the loss of France in his death affects the whole world.

It was in 1900 that, as president of the Royal Astronomical Society, I had the privilege of handing to Poincaré the medal of the Society, and I then attempted to give an appreciation of his work on the theory of the tides, on figures of equilibrium of rotating fluid and on the problem of the three bodies. Again in the preface to the third volume of my collected papers I ventured to describe him as my patron Saint as regards the papers contained in that volume. It brings vividly home to me how great a man he was when I reflect that to one incompetent to appreciate fully one half of his work yet he appears as a star of the first magnitude.

It affords an interesting study to attempt to analyse the difference in the textures of the minds of pure and applied mathematicians. I think that I shall not be doing wrong to the reputation of the psychologists of half a century ago when I say that they thought that when they had successfully analysed the way in which their own minds work they had solved the problem before them. But it was Sir Francis Galton who showed that such a view is erroneous. He pointed out that for many men visual images form the most potent apparatus of thought, but that for others this is not the case. Such visual images are often quaint and illogical, being probably often founded on infantile impressions, but they form the wheels of the clockwork of many minds. The pure geometrician must be a man who is endowed with great powers of visualisation, and this view is confirmed by my recollection of the difficulty of attaining to clear conceptions of the geometry of space until practice in the art of visualisation had enabled one to picture clearly the relationship of lines and surfaces to one another. The pure analyst probably relies far less on visual images, or at least his pictures are not of a geometrical character. I suspect that the mathematician will drift naturally to one branch or another of our science according to the texture of his mind and the nature of the mechanism by which he works.

I wish Galton, who died but recently, could have been here to collect from the great mathematicians now assembled an introspective account of the way in which their minds work. One would like to know whether students of the theory of groups picture to themselves little groups of dots; or are they sheep grazing in a field? Do those who work at the theory of numbers associate colour, or good or bad characters with the lower ordinal numbers, and what are the shapes of the curves in which the successive numbers are arranged? What I have just said will appear pure nonsense to some in this room, others will be recalling what they see, and perhaps some will now for the first time be conscious of their own visual images.

The minds of pure and applied mathematicians probably also tend to differ from one another in the sense of aesthetic beauty. Poincaré has well remarked in his

In this connection I would remark on the extraordinary psychological interest of Poincaré's account, in the chapter from which I have already quoted, of the manner in which he proceeded in attacking a mathematical problem. He describes the unconscious working of the mind, so that his conclusions appeared to his conscious self as revelations from another world. I suspect that we have all been aware of something of the same sort, and like Poincaré have also found that the revelations were not always to be trusted.

Both the pure and the applied mathematician are in search of truth, but the former seeks truth in itself and the latter truths about the universe in which we live. To some men abstract truth has the greater charm, to others the interest in our universe is dominant. In both fields there is room for indefinite advance; but while in pure mathematics every new discovery is a gain, in applied mathematics it is not always easy to find the direction in which progress can be made, because the selection of the conditions essential to the problem presents a preliminary task, and afterwards there arise the purely mathematical difficulties. Thus it appears to me at least, that it is easier to find a field for advantageous research in pure than in applied mathematics. Of course if we regard an investigation in applied mathematics as an exercise in analysis, the correct selection of the essential conditions is immaterial; but if the choice has been wrong the results lose almost all their interest. I may illustrate what I mean by reference to Lord Kelvin's celebrated investigation as to the cooling of the earth. He was not and could not be aware of the radioactivity of the materials of which the earth is formed, and I think it is now generally acknowledged that the conclusions which he deduced as to the age of the earth cannot be maintained; yet the mathematical investigation remains intact.

The appropriate formulation of the problem to be solved is one of the greatest difficulties which beset the applied mathematician, and when he has attained to a true insight but too often there remains the fact that his problem is beyond the reach of mathematical solution. To the layman the problem of the three bodies seems so simple that he is surprised to learn that it cannot be solved completely, and yet we know what prodigies of mathematical skill have been bestowed on it. My own work on the subject cannot be said to involve any such skill at all, unless indeed you describe as skill the procedure of a housebreaker who blows in a safe-door with dynamite instead of picking the lock. It is thus by brute force that this tantalising problem has been compelled to give up some few of its secrets, and great as has been the labour involved I think it has been worth while. Perhaps this work too has done something to encourage others such as Stürmer to similar tasks as in the computation of the orbits of electrons in the neighbourhood of the earth, thus affording an explanation of some of the phenomena of the aurora borealis. To put at their lowest the claims of this clumsy method, which may almost excite the derision of the pure mathematician, it has served to throw light on the celebrated generalisations of Hill and Poincaré.

I appeal then for mercy to the applied mathematician and would ask you to consider in a kindly spirit the difficulties under which he labours. If our methods are often wanting in elegance and do but little to satisfy that aesthetic sense of which I spoke before, yet they are honest attempts to unravel the secrets of the universe in which we live.

We are met here to consider mathematical science in all its branches. Specialisation has become a necessity of modern work and the intercourse which will take place between us in the course of this week will serve to promote some measure of comprehension of the work which is being carried on in other fields than our own. The papers and lectures which you will hear will serve towards this end, but perhaps the personal conversations outside the regular meetings may prove even more useful.

Mr R F Scott, Vice-Chancellor of the University of Cambridge, spoke as follows:The Science of Mathematics is now so wide and is already so much specialised that it may be doubted whether there exists today any man fully competent to understand mathematical research in all its many diverse branches. I, at least, feel how profoundly ill-equipped I am to represent our Society as regards all that vast field of knowledge which we classify as pure mathematics. I must tell you frankly that when I gaze on some of the papers written by men in this room I feel myself much in the same position as if they were written in Sanskrit.

But if there is any place in the world in which so one-sided a President of the body which has the honour to bid you welcome is not wholly out of place it is perhaps Cambridge. It is true that there have been in the past at Cambridge great pure mathematicians such as Cayley and Sylvester, but we surely may claim without undue boasting that our University has played a conspicuous part in the advance of applied mathematics. Newton was as a glory to all mankind, yet we Cambridge men are proud that fate ordained that he should have been Lucasian Professor here. But as regards the part played by Cambridge I refer rather to the men of the last hundred years, such as Airy, Adams, Maxwell, Stokes, Kelvin, and other lesser lights, who have marked out the lines of research in applied mathematics as studied in this University. Then too there are others such as our Chancellor, Lord Rayleigh, who are happily still with us.

Up to a few weeks ago there was one man who alone of all mathematicians might have occupied the place which I hold without misgivings as to his fitness; I mean Henri Poincaré. It was at Rome just four years ago that the first dark shadow fell on us of that illness which has now terminated so fatally. You all remember the dismay which fell on us when the word passed from man to man "Poincaré is ill." We had hoped that we might again have heard from his mouth some such luminous address as that which he gave at Rome; but it was not to be, and the loss of France in his death affects the whole world.

It was in 1900 that, as president of the Royal Astronomical Society, I had the privilege of handing to Poincaré the medal of the Society, and I then attempted to give an appreciation of his work on the theory of the tides, on figures of equilibrium of rotating fluid and on the problem of the three bodies. Again in the preface to the third volume of my collected papers I ventured to describe him as my patron Saint as regards the papers contained in that volume. It brings vividly home to me how great a man he was when I reflect that to one incompetent to appreciate fully one half of his work yet he appears as a star of the first magnitude.

It affords an interesting study to attempt to analyse the difference in the textures of the minds of pure and applied mathematicians. I think that I shall not be doing wrong to the reputation of the psychologists of half a century ago when I say that they thought that when they had successfully analysed the way in which their own minds work they had solved the problem before them. But it was Sir Francis Galton who showed that such a view is erroneous. He pointed out that for many men visual images form the most potent apparatus of thought, but that for others this is not the case. Such visual images are often quaint and illogical, being probably often founded on infantile impressions, but they form the wheels of the clockwork of many minds. The pure geometrician must be a man who is endowed with great powers of visualisation, and this view is confirmed by my recollection of the difficulty of attaining to clear conceptions of the geometry of space until practice in the art of visualisation had enabled one to picture clearly the relationship of lines and surfaces to one another. The pure analyst probably relies far less on visual images, or at least his pictures are not of a geometrical character. I suspect that the mathematician will drift naturally to one branch or another of our science according to the texture of his mind and the nature of the mechanism by which he works.

I wish Galton, who died but recently, could have been here to collect from the great mathematicians now assembled an introspective account of the way in which their minds work. One would like to know whether students of the theory of groups picture to themselves little groups of dots; or are they sheep grazing in a field? Do those who work at the theory of numbers associate colour, or good or bad characters with the lower ordinal numbers, and what are the shapes of the curves in which the successive numbers are arranged? What I have just said will appear pure nonsense to some in this room, others will be recalling what they see, and perhaps some will now for the first time be conscious of their own visual images.

The minds of pure and applied mathematicians probably also tend to differ from one another in the sense of aesthetic beauty. Poincaré has well remarked in his

*Science et Méthode*:And again he writes:It is surprising to see sensitivity invoked in connection with mathematical demonstrations which, it seems, can only interest intelligence. It would be to forget the feeling of mathematical beauty, of the harmony of numbers and shapes, of geometric elegance. It is a true aesthetic feeling that all true mathematicians know. And this is sensitivity.

Of course there is every gradation from one class of mind to the other, and in some the aesthetic sense is dominant and in others subordinate.Useful combinations are precisely the most beautiful, I mean the ones that can best charm this special sensitivity that all mathematicians know, but which laymen ignore so much that they are often tempted to smile at it.

In this connection I would remark on the extraordinary psychological interest of Poincaré's account, in the chapter from which I have already quoted, of the manner in which he proceeded in attacking a mathematical problem. He describes the unconscious working of the mind, so that his conclusions appeared to his conscious self as revelations from another world. I suspect that we have all been aware of something of the same sort, and like Poincaré have also found that the revelations were not always to be trusted.

Both the pure and the applied mathematician are in search of truth, but the former seeks truth in itself and the latter truths about the universe in which we live. To some men abstract truth has the greater charm, to others the interest in our universe is dominant. In both fields there is room for indefinite advance; but while in pure mathematics every new discovery is a gain, in applied mathematics it is not always easy to find the direction in which progress can be made, because the selection of the conditions essential to the problem presents a preliminary task, and afterwards there arise the purely mathematical difficulties. Thus it appears to me at least, that it is easier to find a field for advantageous research in pure than in applied mathematics. Of course if we regard an investigation in applied mathematics as an exercise in analysis, the correct selection of the essential conditions is immaterial; but if the choice has been wrong the results lose almost all their interest. I may illustrate what I mean by reference to Lord Kelvin's celebrated investigation as to the cooling of the earth. He was not and could not be aware of the radioactivity of the materials of which the earth is formed, and I think it is now generally acknowledged that the conclusions which he deduced as to the age of the earth cannot be maintained; yet the mathematical investigation remains intact.

The appropriate formulation of the problem to be solved is one of the greatest difficulties which beset the applied mathematician, and when he has attained to a true insight but too often there remains the fact that his problem is beyond the reach of mathematical solution. To the layman the problem of the three bodies seems so simple that he is surprised to learn that it cannot be solved completely, and yet we know what prodigies of mathematical skill have been bestowed on it. My own work on the subject cannot be said to involve any such skill at all, unless indeed you describe as skill the procedure of a housebreaker who blows in a safe-door with dynamite instead of picking the lock. It is thus by brute force that this tantalising problem has been compelled to give up some few of its secrets, and great as has been the labour involved I think it has been worth while. Perhaps this work too has done something to encourage others such as Stürmer to similar tasks as in the computation of the orbits of electrons in the neighbourhood of the earth, thus affording an explanation of some of the phenomena of the aurora borealis. To put at their lowest the claims of this clumsy method, which may almost excite the derision of the pure mathematician, it has served to throw light on the celebrated generalisations of Hill and Poincaré.

I appeal then for mercy to the applied mathematician and would ask you to consider in a kindly spirit the difficulties under which he labours. If our methods are often wanting in elegance and do but little to satisfy that aesthetic sense of which I spoke before, yet they are honest attempts to unravel the secrets of the universe in which we live.

We are met here to consider mathematical science in all its branches. Specialisation has become a necessity of modern work and the intercourse which will take place between us in the course of this week will serve to promote some measure of comprehension of the work which is being carried on in other fields than our own. The papers and lectures which you will hear will serve towards this end, but perhaps the personal conversations outside the regular meetings may prove even more useful.

Gentlemen, It is my privilege today on behalf of the University of Cambridge and its Colleges to offer to Members of the Congress a hearty welcome from the resident body.

Sir George Darwin has dwelt on the more serious aspects of the meeting and work of the Congress, may I express the hope that it will also have its lighter and more personal side? That we shall all have the privilege and pleasure of making the personal acquaintance of many well known to us both by name and by fame, and that those of our visitors who are not familiar with the College life of Oxford and Cambridge will learn something of a feature so distinctive of the two ancient English Universities. If the Congress comes at a time when it is not possible to see the great body of our students either at work or at play, the choice of date at least renders it possible that many of our visitors may enjoy for a time that Collegiate life which has so many attractions.

I see that one of the Sections of the Congress deals with historical and didactical questions. Those members of the Congress who are interested in these subjects will have an opportunity of learning on the spot something of our methods in Cambridge, and of the history of our chief Mathematical Examination, the Mathematical Tripos, and of its influence on the study and progress of Mathematics both in Cambridge and Great Britain.

The researches of Dr Venn seem to point to the fact that until it was altered at a very recent date the Mathematical Tripos represented something like the oldest example in Europe of a competitive Examination with an order of merit. Those who are interested in such matters of history will find much to interest them in Mr Rouse Ball's

Many Cambridge mathematicians, as the names given by Sir George Darwin testify, studied mathematics for its own sake and with the view of extending the boundaries of knowledge. Many others, probably the great majority, studied mathematics with their eyes fixed upon the Mathematical Tripos, with the view in the first place of being examined and afterwards of acting as examiner in it. The tendency at Cambridge has been to give great minuteness to the study of any particular branch of mathematics. To stimulate the invention of what we call "Problems," examples of more general theories. If I may borrow a simile from the study of Literature the tendency was to produce critics and editors rather than authors or men of letters, followers rather than investigators. The effect must I think be obvious to any one who compares Cambridge Text Books and Treatises with those of the Continental Schools of Mathematics. I may illustrate what I mean by referring to the

Professor Ernest W Hobson, Senior Secretary of the Organising Committee, stated that the number of persons who had joined the Congress up to 22.00 on Wednesday, 21 August, was 670, the number of representatives of different countries being as follows: Argentine 4, Austria 19, Belgium 4, Bulgaria 1, Canada 4, Chile 1, Denmark 5, Egypt 2, France 42, Germany 70, Great Britain 250, Greece 5, Holland 9, Hungary 19, India 3, Italy 38, Japan 3, Mexico 1, Norway 4, Portugal 3, Romania 5, Russia 38, Serbia 1, Spain 25, Sweden 13, Switzerland 9, United States 82. He also called the attention of the Members of the Congress to the exhibition of books, models and machines (chiefly calculating machines) arranged in two rooms of the Cavendish Laboratory.
Sir George Darwin has dwelt on the more serious aspects of the meeting and work of the Congress, may I express the hope that it will also have its lighter and more personal side? That we shall all have the privilege and pleasure of making the personal acquaintance of many well known to us both by name and by fame, and that those of our visitors who are not familiar with the College life of Oxford and Cambridge will learn something of a feature so distinctive of the two ancient English Universities. If the Congress comes at a time when it is not possible to see the great body of our students either at work or at play, the choice of date at least renders it possible that many of our visitors may enjoy for a time that Collegiate life which has so many attractions.

I see that one of the Sections of the Congress deals with historical and didactical questions. Those members of the Congress who are interested in these subjects will have an opportunity of learning on the spot something of our methods in Cambridge, and of the history of our chief Mathematical Examination, the Mathematical Tripos, and of its influence on the study and progress of Mathematics both in Cambridge and Great Britain.

The researches of Dr Venn seem to point to the fact that until it was altered at a very recent date the Mathematical Tripos represented something like the oldest example in Europe of a competitive Examination with an order of merit. Those who are interested in such matters of history will find much to interest them in Mr Rouse Ball's

*History of the Study of Mathematics at Cambridge*. The subject is to me and I hope to others an interesting one. There can be no doubt that the Examination and the preparation for it has had a profound influence on Mathematical studies at Cambridge.Many Cambridge mathematicians, as the names given by Sir George Darwin testify, studied mathematics for its own sake and with the view of extending the boundaries of knowledge. Many others, probably the great majority, studied mathematics with their eyes fixed upon the Mathematical Tripos, with the view in the first place of being examined and afterwards of acting as examiner in it. The tendency at Cambridge has been to give great minuteness to the study of any particular branch of mathematics. To stimulate the invention of what we call "Problems," examples of more general theories. If I may borrow a simile from the study of Literature the tendency was to produce critics and editors rather than authors or men of letters, followers rather than investigators. The effect must I think be obvious to any one who compares Cambridge Text Books and Treatises with those of the Continental Schools of Mathematics. I may illustrate what I mean by referring to the

*Mathematical Problems*of the late Mr Joseph Wolstenholme, a form of work I believe without a parallel in the mathematical literature of other nations. The fashion is fading away, but while you are in Cambridge I commend it to your notice.The first general meeting of the Congress was held at 14.30.

On the motion of Professor Mittag-Leffler, seconded by Professor Enriques, Sir G H Darwin was elected President of the Congress.

On the motion of the President it was agreed that Lord Rayleigh be made Honorary President of the Congress (Président d'honneur).

On the motion of the President, Vice-Presidents of the Congress were elected as follows:- W von Dyck, L Fejér, R Fujisawa, J Hadamard, J L W V Jensen, P A MacMahon, G Mittag-Leffler, E H Moore, F Rudio, P H Schoute, M S Smoluchowski, V A Steklov, V Volterra.

On the motion of the President, General Secretaries of the Congress were elected as follows:- E W Hobson, A E H Love.

Sir George Greenhill made the following statement in regard to the work of the International Commission on the Teaching of Mathematics:

The statement I have to make, Sir, to the Congress, is given in the formal words following:

1. The International Commission on the Teaching of Mathematics was appointed at the Rome Congress, on the recommendation of the Members of Section IV.

2. The several countries, in one way or another, have recognised officially the work, and have contributed financial support.

3. About 150 reports have been published, and about 50 more will appear later.

4. The Commission will report in certain Sessions of Section IV.

5. The Commission hopes to be continued in power, in order that the work now in progress may be brought to completion. A Resolution to this effect will be offered at the final Meeting of the Congress.

At 15.30 Professor F Enriques delivered his lecture 1. The International Commission on the Teaching of Mathematics was appointed at the Rome Congress, on the recommendation of the Members of Section IV.

2. The several countries, in one way or another, have recognised officially the work, and have contributed financial support.

3. About 150 reports have been published, and about 50 more will appear later.

4. The Commission will report in certain Sessions of Section IV.

5. The Commission hopes to be continued in power, in order that the work now in progress may be brought to completion. A Resolution to this effect will be offered at the final Meeting of the Congress.

*Il significato della critica dei principii nello sviluppo delle matematiche.*

At 17.00 Professor E W Brown delivered his lecture

*Periodicities in the Solar System.*

At 21.00 Section IV met.

*Friday,*23

*August*.

The various Sections met at 9.30.

At 15.30 Professor Edmund Landau delivered his lecture

*Gelöste und ungelöste Probleme aus der Theorie der Primzahl Verteilung und der Riemannschen Zetafunktion.*

At 17.00 Prince Boris Borisovich Galitzin delivered his lecture

*The principles of instrumental seismology.*

At 21.00 the Members of the Congress were received at a conversazione in the Fitzwilliam Museum by the Chancellor of the University, The Rt Hon Lord Rayleigh, O.M.

*Saturday,*24

*August*.

The various Sections met at 9.30.

At 15.30 Professor É Borel delivered his lecture

*Définition et domaine d'existence des fonctions monogènes uniformes.*

At 17.00 Sir William Henry White delivered his lecture

*The place of mathematics in engineering practice.*

*Sunday,*25

*August*.

At 15.00 the Members of the Congress were received at an afternoon party in the Garden of Christ's College by the President of the Congress, Sir G H Darwin. At 21.00 the Members of the Congress were invited to be present at an Organ Recital in the Chapel of King's College.

*Monday,*26

*August*.

The various Sections met at 9.30.

A meeting of Section IV (b) was held at 15.00.

A meeting of Section I was held at 15.30.

In the afternoon Members of the Congress made an excursion to Ely and visited the Cathedral. Other Members visited the works of the Cambridge Scientific Instrument Company and the University Observatory.

At 21.00 the Members of the Congress were entertained in the Hall and Cloisters of Trinity College by the Master and Fellows of the College.

*Tuesday,*27

*August*.

The various Sections met at 9.30.

At 15.30 Professor Maxime Bôcher delivered his lecture

*Boundary problems in one dimension.*

At 17.00 Sir Joseph Larmor delivered his lecture

*The dynamics of radiation.*

In the afternoon a number of Members of the Congress proceeded to the Mill Road Cemetery for the purpose of depositing a wreath upon the grave of the late Professor Arthur Cayley. An address was delivered by Professor Samuel Dickstein. Other Members visited the works of the Cambridge Scientific Instrument Company.

**3. Closing session of the Congress.**

At 21.00 the final meeting of the Congress was held.

The President Sir G H Darwin read a telegram from Professor Vito Volterra regretting that family reasons prevented his attending the Congress, and proceeded to speak as follows:

Ladies and Gentlemen,

We meet tonight for the final Conference of the present Congress. The majority of those present in the room will have been aware that a procession was formed today to lay a wreath on the tomb of Cayley. This has touched the hearts of our University. It had, I believe, been suggested that a more permanent wreath should have been deposited; but no such things can be obtained at short notice, and the final arrangement adopted is that a silver wreath shall be made and presented to the University, whose authorities will I am sure gratefully accept it and deposit it in some appropriate place, where it will remain as a permanent memorial of the recognition accorded by the mathematicians of all nations to our great investigator. The subscribers have entrusted the carrying out of this to our Organising Committee in Cambridge.

I will now explain the order in which we think that it will be convenient for us to carry out our business of tonight. At the last meeting at Rome various resolutions were adopted, and I shall draw your attention to all of them which may give rise to any further discussion tonight, and I shall then successively call on speakers who may have resolutions to propose. In doing this the order of subjects will be followed as I find them in the procès-verbal of the Roman Congress. After these matters are decided, opportunity will of course be afforded to any of our members who may have new subjects on which they have proposals to make.

The first resolution at Rome concerned the work of the International Commission on the Teaching of Mathematics, and a resolution will be proposed as to this. The second resolution was one as to the unification of vectorial notations. I learn from Jacques Hadamard that exchange of views has taken place on this subject during the last four years, but that it has not been found possible to arrive at any definite conclusions. No resolution will be proposed on the present occasion, but it is hoped that by the time the next Congress takes place something may have been achieved and that the matter will be brought forward again.

It was proposed at Rome that a constitution should be formed for an International Association of Mathematicians. I have not heard that any proposal will be made tonight and I do not hesitate to express my own opinion that our existing arrangements for periodical Congresses meet the requirements of the case better than would a permanent organisation of the kind suggested.

There has been a resolution as to the improvement and unification of the methods of pure and applied mathematics. This subject seems to be sufficiently taken cognisance of as part of the work of the Commission on Teaching, and I cannot think that any further action on our part is needed.

Next there followed a resolution as to the publication of the works of Euler, and a resolution as to this will be proposed tonight.

In this connection I would remark that a complete edition of the works of the immortal Herschel is in course of publication by the Royal Society.

An important matter has to be determined tonight, namely the choice of the place and of the time of the next Congress, and I shall call on Professor Mittag-Leffler to speak to this subject, and of course others may also speak if they desire.

Opportunity will then be afforded to any others who may have proposals to bring forward. When these matters of business are decided I shall say a few words as to the Congress which is now terminating.

The following resolution was moved by Mr C Godfrey, seconded by Professor Walther von Dyck and carried nem. con.:We meet tonight for the final Conference of the present Congress. The majority of those present in the room will have been aware that a procession was formed today to lay a wreath on the tomb of Cayley. This has touched the hearts of our University. It had, I believe, been suggested that a more permanent wreath should have been deposited; but no such things can be obtained at short notice, and the final arrangement adopted is that a silver wreath shall be made and presented to the University, whose authorities will I am sure gratefully accept it and deposit it in some appropriate place, where it will remain as a permanent memorial of the recognition accorded by the mathematicians of all nations to our great investigator. The subscribers have entrusted the carrying out of this to our Organising Committee in Cambridge.

I will now explain the order in which we think that it will be convenient for us to carry out our business of tonight. At the last meeting at Rome various resolutions were adopted, and I shall draw your attention to all of them which may give rise to any further discussion tonight, and I shall then successively call on speakers who may have resolutions to propose. In doing this the order of subjects will be followed as I find them in the procès-verbal of the Roman Congress. After these matters are decided, opportunity will of course be afforded to any of our members who may have new subjects on which they have proposals to make.

The first resolution at Rome concerned the work of the International Commission on the Teaching of Mathematics, and a resolution will be proposed as to this. The second resolution was one as to the unification of vectorial notations. I learn from Jacques Hadamard that exchange of views has taken place on this subject during the last four years, but that it has not been found possible to arrive at any definite conclusions. No resolution will be proposed on the present occasion, but it is hoped that by the time the next Congress takes place something may have been achieved and that the matter will be brought forward again.

It was proposed at Rome that a constitution should be formed for an International Association of Mathematicians. I have not heard that any proposal will be made tonight and I do not hesitate to express my own opinion that our existing arrangements for periodical Congresses meet the requirements of the case better than would a permanent organisation of the kind suggested.

There has been a resolution as to the improvement and unification of the methods of pure and applied mathematics. This subject seems to be sufficiently taken cognisance of as part of the work of the Commission on Teaching, and I cannot think that any further action on our part is needed.

Next there followed a resolution as to the publication of the works of Euler, and a resolution as to this will be proposed tonight.

In this connection I would remark that a complete edition of the works of the immortal Herschel is in course of publication by the Royal Society.

An important matter has to be determined tonight, namely the choice of the place and of the time of the next Congress, and I shall call on Professor Mittag-Leffler to speak to this subject, and of course others may also speak if they desire.

Opportunity will then be afforded to any others who may have proposals to bring forward. When these matters of business are decided I shall say a few words as to the Congress which is now terminating.

That the Congress expresses its appreciation of the support given to its Commission on the Teaching of Mathematics by various governments, institutions, and individuals; that the Central Committee composed of F Klein (Göttingen), Sir G Greenhill (London) and H Fehr (Geneva) be continued in power and that, at its request, David Eugene Smith of New York be added to its number; that the Delegates be requested to continue their good offices in securing the cooperation of their respective governments, and in carrying on the work; and that the Commission be requested to make such further report at the Sixth International Congress, and to hold such conferences in the meantime, as the circumstances warrant.

A French translation of the resolution was read to the meeting by M Bioche. In seconding the resolution Professor von Dyck spoke as follows:
I wish to second the motion, but in doing so you will allow me to insist with a few words upon the prominent work done by the International Committee on Teaching of Mathematics during these last four years. None of us who were present in Rome could even imagine what an immense labour was to be undertaken when Dr D E Smith proposed a comparative investigation on mathematical teaching.

Now, by the activity of the splendid organisation of the Central Committee, under the guidance of Klein, Greenhill, and Fehr, with the worthy help of D E Smith, every country in nearly every part of the world has contributed in its own department to the Reports for Cambridge - so that there were about 150 different volumes with about 300 articles brought before the Congress - papers which were not all read but were aptly spoken about by the collaborators.

So we will congratulate the Committee upon the work already done, and we have to express our most hearty thanks both to the Central and Local Committees and the collaborators.

But furthermore we have to congratulate ourselves that this Committee will remain still in charge and will continue and finish the work.

For us, the outsiders, the series of reports is like a series of a very large number of coefficients to be calculated. And the problem arises now to find the principles under which they may be grouped and compared with each other according to their individuality and their quality. Whom could we better entrust with that problem than this acting committee, which has been at work all this time, and to whom we are even now so deeply obliged?

The following resolution was moved by Professor August Gutzmer and carried:Now, by the activity of the splendid organisation of the Central Committee, under the guidance of Klein, Greenhill, and Fehr, with the worthy help of D E Smith, every country in nearly every part of the world has contributed in its own department to the Reports for Cambridge - so that there were about 150 different volumes with about 300 articles brought before the Congress - papers which were not all read but were aptly spoken about by the collaborators.

So we will congratulate the Committee upon the work already done, and we have to express our most hearty thanks both to the Central and Local Committees and the collaborators.

But furthermore we have to congratulate ourselves that this Committee will remain still in charge and will continue and finish the work.

For us, the outsiders, the series of reports is like a series of a very large number of coefficients to be calculated. And the problem arises now to find the principles under which they may be grouped and compared with each other according to their individuality and their quality. Whom could we better entrust with that problem than this acting committee, which has been at work all this time, and to whom we are even now so deeply obliged?

In accordance with a wash that has been repeatedly expressed by successive International Congresses of Mathematicians, and in particular, in accordance with the resolution adopted at Rome, concerning the publication of the collected works of Leonhard Euler, the fifth International Congress of Mathematicians, assembled at Cambridge, expresses its warmest thanks to the Schweizerische Naturforschende Gesellschaft for their efforts in inaugurating the great work, and for the magnificent style in which the five volumes already published have been completed. The Congress expresses the hope that the scientific world will continue to exhibit that sustained interest in the undertaking which it has hitherto shown.

Professor G Mittag-Leffler presented an invitation to the Congress to hold its next meeting at Stockholm in 1916. The following is the text of the invitation:
On behalf of the members of the third class of the Royal Swedish Academy of Sciences, on behalf of the Swedish editorial staff of the journal Acta Mathematica and all the Swedish mathematicians, I have the honour to invite the International Congress of Mathematicians to meet in Stockholm in the year 1916.

Our august sovereign King Gustave has graciously entrusted me with the task of expressing to the Congress that he would welcome it with pleasure in his capital and that he would be ready to take it under his patronage during its stay in Stockholm.

The rest of us would consider ourselves very happy if the Congress wanted to accept our invitation, and we will do everything in our power to make the members' stay in our country as pleasant and informative as possible.

Professor Emanuel Beke presented an invitation to the Congress to hold its meeting of 1920 at Budapest. The following is the text of the invitation:Our august sovereign King Gustave has graciously entrusted me with the task of expressing to the Congress that he would welcome it with pleasure in his capital and that he would be ready to take it under his patronage during its stay in Stockholm.

The rest of us would consider ourselves very happy if the Congress wanted to accept our invitation, and we will do everything in our power to make the members' stay in our country as pleasant and informative as possible.

On behalf of Hungarian mathematicians, I have the honour to invite the Seventh International Congress to come and take place in 1920 in Budapest, the capital of Hungary.

While knowing that it is the Stockholm Congress that will have to decide on our invitation, we are already presenting it here, in accordance with an excellent practice adopted by previous Congresses.

I am authorised to announce to you that the competent scientific institutions as well as the royal Hungarian government will give us their effective assistance and all their support.

The homeland of the Bolyai and its beautiful capital will be proud to be able to offer their hospitality to the learned representatives of the mathematical sciences, who will want to honour us with their presence.

On the motion of the President the invitation to Stockholm was accepted nem. con. The President stated that the Congress noted with gratitude the invitation to Budapest, but the decision as to the place of the next meeting after that of 1916 would properly be made at Stockholm.
While knowing that it is the Stockholm Congress that will have to decide on our invitation, we are already presenting it here, in accordance with an excellent practice adopted by previous Congresses.

I am authorised to announce to you that the competent scientific institutions as well as the royal Hungarian government will give us their effective assistance and all their support.

The homeland of the Bolyai and its beautiful capital will be proud to be able to offer their hospitality to the learned representatives of the mathematical sciences, who will want to honour us with their presence.

Professor Cyparissos Stephanos expressed the hope that the Congress would meet in Athens in 1920 or 1924.

It was resolved that the following telegram be sent to Lord Rayleigh:

The President then spoke as follows:The Fifth International Congress of Mathematicians at the end of its work addresses to the illustrious Chancellor of the University of Cambridge, to the great creator in the fields of mathematical and physical sciences, respectful expression of his tributes and his admiration.

We have come to the end of a busy week, and I have the impression that the papers and lectures which you have heard have been worthy of the occasion. I trust too that you will look back on the meeting as a week of varied interests. The weather has been such that in a more superstitious age we should surely have concluded that heaven did not approve of our efforts; but fortunately today we regard it rather as a matter for the consideration of Section III (a) to decide why it is that solar radiation acting on a layer of compressible fluid on the planet should have selected England as the seat of its most unkindly efforts in the way of precipitation. Notwithstanding this I cannot think that I have wholly misinterpreted the looks and the words of those of whom I have seen so much during these latter days, when I express the conviction that you have enjoyed yourselves. There is much of the middle ages in our old Colleges at Cambridge, and it is only at Oxford that you can find any parallel to what you have seen here. Many of you will have the opportunity tomorrow under the guidance of Professor Love of seeing the wonderful beauties of Oxford, and I express the hope that the weather may be such as to make us Cambridge men jealous of the good fortune of Oxford.

I feel assured that all of you must realise how long and arduous are the preparations for such a Congress as this. I believe that the arrangements made for your reception have been on the whole satisfactory, and I wish to tell you how much you owe in this respect to Professor Hobson. For months past he has been endeavouring to do all that was in his power to render this meeting both efficient and agreeable. During the last few weeks he has been joined by Professor Love from Oxford, and they have both been busy from morning to night at countless matters which needed decision. You are perhaps aware that our Parliament in its wisdom has decided that coal-miners shall not be allowed to work for more than eight hours a day. There has been no eight hours bill for the Secretaries of this Congress, and if I were to specify a time for the work of Hobson and Love I should put it at sixteen hours a day. As President of the Organising Committee and subsequently of the Congress I wish to express my warm thanks to them for all that they have done. Before closing the meeting I shall ask them to say a few words, and Professor Hobson will take this opportunity of telling you something as to the final numbers of those attending.

Sir Joseph Larmor has undertaken the financial side of our work. His work although less arduous than that of the Secretaries has been not less responsible.

Each department of the social arrangements has been in the charge of some one man, and I want to thank them all for what they have done. Mr Hinks kindly served me as my special aide-de-camp, and has also been of inestimable service to Lady Darwin and the Committee of Ladies in entertaining the ladies who are present here. May I be pardoned if I say that I think the reception by the Chancellor and Lady Rayleigh at the Fitzwilliam Museum was a brilliant one, and I think you should know that every detail was carried out at the suggestion and under the care of Mr Hinks. His work was not facilitated by the fact that a number of things had to be changed at the last minute on account of the bad weather, but I doubt whether any traces of the changes made will have struck you.

Then I desire also to express our warm thanks to Mr A W Smith who was in charge of the reception room which has proved so convenient an institution. He had, as Assistant Secretary, countless other matters to which to attend and he has carried out all these with the highest success.

Finally I am sure that I may take on myself as your President to express to the Authorities of the University our gratitude for the use of these rooms for the meeting, and to the Committee (called by us the Syndicate) of the Fitzwilliam Museum, responsible for many valuable collections, for the loan to the Chancellor of the Museum for our reception. We also desire to acknowledge the pleasure we had in the beautiful reception given to us by the Master and Fellows of Trinity College, to the Master in person for the interesting lecture which he gave to the ladies, and to Colonel Harding and Sir G Waldstein for their kindness in receiving the ladies at their country houses.

Professor Hobson stated the number of Members of the Congress as follows:I feel assured that all of you must realise how long and arduous are the preparations for such a Congress as this. I believe that the arrangements made for your reception have been on the whole satisfactory, and I wish to tell you how much you owe in this respect to Professor Hobson. For months past he has been endeavouring to do all that was in his power to render this meeting both efficient and agreeable. During the last few weeks he has been joined by Professor Love from Oxford, and they have both been busy from morning to night at countless matters which needed decision. You are perhaps aware that our Parliament in its wisdom has decided that coal-miners shall not be allowed to work for more than eight hours a day. There has been no eight hours bill for the Secretaries of this Congress, and if I were to specify a time for the work of Hobson and Love I should put it at sixteen hours a day. As President of the Organising Committee and subsequently of the Congress I wish to express my warm thanks to them for all that they have done. Before closing the meeting I shall ask them to say a few words, and Professor Hobson will take this opportunity of telling you something as to the final numbers of those attending.

Sir Joseph Larmor has undertaken the financial side of our work. His work although less arduous than that of the Secretaries has been not less responsible.

Each department of the social arrangements has been in the charge of some one man, and I want to thank them all for what they have done. Mr Hinks kindly served me as my special aide-de-camp, and has also been of inestimable service to Lady Darwin and the Committee of Ladies in entertaining the ladies who are present here. May I be pardoned if I say that I think the reception by the Chancellor and Lady Rayleigh at the Fitzwilliam Museum was a brilliant one, and I think you should know that every detail was carried out at the suggestion and under the care of Mr Hinks. His work was not facilitated by the fact that a number of things had to be changed at the last minute on account of the bad weather, but I doubt whether any traces of the changes made will have struck you.

Then I desire also to express our warm thanks to Mr A W Smith who was in charge of the reception room which has proved so convenient an institution. He had, as Assistant Secretary, countless other matters to which to attend and he has carried out all these with the highest success.

Finally I am sure that I may take on myself as your President to express to the Authorities of the University our gratitude for the use of these rooms for the meeting, and to the Committee (called by us the Syndicate) of the Fitzwilliam Museum, responsible for many valuable collections, for the loan to the Chancellor of the Museum for our reception. We also desire to acknowledge the pleasure we had in the beautiful reception given to us by the Master and Fellows of Trinity College, to the Master in person for the interesting lecture which he gave to the ladies, and to Colonel Harding and Sir G Waldstein for their kindness in receiving the ladies at their country houses.

Number of Members of the Congress 708.

Number of effective Members 574.

He also thanked the Members of the Congress of all nations for their courtesy in their correspondence with the General Secretaries during the time of preparation for the Congress.

Professor G Mittag-Leffler then spoke as follows:Number of effective Members 574.

He also thanked the Members of the Congress of all nations for their courtesy in their correspondence with the General Secretaries during the time of preparation for the Congress.

Ladies and gentlemen,

The foreigners who took part in this Fifth International Congress of Mathematicians which has just ended asked me to be with our English colleagues and hosts the interpreter of their lively and warm gratitude for the charming welcome we have received. It is with particular pleasure that we stayed in this city filled with great scientific memories, the cradle of this illustrious university where ancient customs and modern thought could unite like nowhere else. Thanks to the excellent leadership of the organising committee, the varied and powerful scientific forces that have gathered here have been able to assert themselves in the most fruitful manner. All of us who took part in the work of the Congress received new and abundant suggestions for our own future work.

For mathematics, congresses are perhaps of greater importance than for other sciences. Mathematics, the science of numbers, the science of sciences even when it helps geometric representation, or when it tries to adapt to external experience, however only deals with pure abstractions. This is what makes the study of ideas communicated only by the printed word much more laborious than in the other sciences whose object is rather concrete. It follows that the study of literature in mathematics, perhaps even more than in these other sciences, finds a precious complement in the verbal exchange of ideas. I therefore believe I am interpreting our unanimous wishes by expressing the hope that the Fifth International Congress of Mathematicians is only a term in a never-ending series of similar congresses, renewed every four years.

I also think I am expressing everyone's thought by saying that the admirable way in which the organising committee of the Cambridge Congress was able to prepare our meetings will be a model for us in the future. It is to our illustrious President Sir George Darwin, to our tireless secretaries gifted with this practical sense that we recognise in the English, Professors Hobson and Love, to Treasurer Sir J Larmor and to the other members of the committee that we owe this result. We thank them wholeheartedly by assuring them that our stay here will be for each of us an unforgettable memory.

The foreigners who took part in this Fifth International Congress of Mathematicians which has just ended asked me to be with our English colleagues and hosts the interpreter of their lively and warm gratitude for the charming welcome we have received. It is with particular pleasure that we stayed in this city filled with great scientific memories, the cradle of this illustrious university where ancient customs and modern thought could unite like nowhere else. Thanks to the excellent leadership of the organising committee, the varied and powerful scientific forces that have gathered here have been able to assert themselves in the most fruitful manner. All of us who took part in the work of the Congress received new and abundant suggestions for our own future work.

For mathematics, congresses are perhaps of greater importance than for other sciences. Mathematics, the science of numbers, the science of sciences even when it helps geometric representation, or when it tries to adapt to external experience, however only deals with pure abstractions. This is what makes the study of ideas communicated only by the printed word much more laborious than in the other sciences whose object is rather concrete. It follows that the study of literature in mathematics, perhaps even more than in these other sciences, finds a precious complement in the verbal exchange of ideas. I therefore believe I am interpreting our unanimous wishes by expressing the hope that the Fifth International Congress of Mathematicians is only a term in a never-ending series of similar congresses, renewed every four years.

I also think I am expressing everyone's thought by saying that the admirable way in which the organising committee of the Cambridge Congress was able to prepare our meetings will be a model for us in the future. It is to our illustrious President Sir George Darwin, to our tireless secretaries gifted with this practical sense that we recognise in the English, Professors Hobson and Love, to Treasurer Sir J Larmor and to the other members of the committee that we owe this result. We thank them wholeheartedly by assuring them that our stay here will be for each of us an unforgettable memory.

*Wednesday,*28

*August*.

Members of the Congress took part in an excursion to Oxford, and a reception at Hatfield House, on the invitation of Lord Salisbury.

**4. Opening remarks by Section Chairmen.**

**1. Section I. Arithmetic, Algebra, Analysis.**

The Section met at 9.30. Professor Edwin Bailey Elliott was in the Chair. Dr Thomas John l'Anson Bromwich was elected Secretary, and Professor I Bendixson and Professor J C Fields were elected Assistant Secretaries. Professor E Landau was elected Chairman for Saturday, 24 August.

The Chairman addressed the Section as follows:

Ladies and Gentlemen,

It is a great honour to be asked to preside today over this Section I of the Fifth International Congress of Mathematicians. My first pleasant duty is to address a few words of welcome to the Section. The many distinguished Analysts and Arithmeticians who are now honouring their fellow-workers in Cambridge by their presence will I trust carry back to their own countries pleasant memories of their stay in this famous university town, and a strong sense of the vitality of Mathematical investigation in its English home. On the other side they will I am sure leave here lasting memories of good work done in promoting combined effort for the advancement of Mathematics. There was a time not long ago when British Mathematicians may have been thought too self-centred. If the judgment were ever correct, it is so no longer. We are alive to what is being done elsewhere, and now aim at cooperation. Our Academical methods are being modified. The severity of examination competition has been relaxed. We shall not give it up entirely: for we have to think not only of providing proficient mathematicians, but also of using mathematical training for the development of men, for the cultivation of exactness of thought, and the power of grasping situations and dealing with problems that arise in life. But the furtherance of mathematical thought is no longer secondary. Perhaps in times past we were too much occupied with exercises of skill and ingenuity, too much, some of us, engrossed with the struggle for absolute perfection in the mastery of limited curricula, too serenely contented with the older analysis. But our slowness in assimilating new ideas by adequate study of the writings of the great masters of the newer analysis is over. Here as elsewhere a younger mathematician now realises that, after grounding himself in common knowledge, he must choose his department of higher study, must acquaint himself by prolonged effort with original authorities, and then produce for himself. A Congress like this will, I feel sure, greatly assist those who are inculcating the sound doctrine.

There is one special reason why I value the opportunity which has been given me of saying these few words of welcome. Like yourselves, ladies and gentlemen from abroad, I am a guest at Cambridge. Not being a Cambridge man, I can let myself say that, while Cambridge has had the honour of inviting you and has the pleasure of entertaining you, the welcome is extended to you by all the mathematicians of the United Kingdom. Our other universities, and my own of Oxford in particular, have Mathematical Faculties of which they are proud, and they do not sink their individualities in their consciousness of the greatness of that of Cambridge. But we owe many times more to Cambridge in the domain of mathematics than Cambridge owes to us. We come here for inspiration, and not infrequently for men. The small band of Oxonians here now will go back grateful to their Cambridge hosts, and yours, for one more benefit conferred in the opportunity of meeting you.

I will delay you no longer, ladies and gentlemen, except to refer in one word to the unspeakable loss sustained in the death of Henri Poincaré, whom in this Section we think of as the prince of analysts, and whom we had hoped to see here today. A higher power has ordered otherwise.

It is a great honour to be asked to preside today over this Section I of the Fifth International Congress of Mathematicians. My first pleasant duty is to address a few words of welcome to the Section. The many distinguished Analysts and Arithmeticians who are now honouring their fellow-workers in Cambridge by their presence will I trust carry back to their own countries pleasant memories of their stay in this famous university town, and a strong sense of the vitality of Mathematical investigation in its English home. On the other side they will I am sure leave here lasting memories of good work done in promoting combined effort for the advancement of Mathematics. There was a time not long ago when British Mathematicians may have been thought too self-centred. If the judgment were ever correct, it is so no longer. We are alive to what is being done elsewhere, and now aim at cooperation. Our Academical methods are being modified. The severity of examination competition has been relaxed. We shall not give it up entirely: for we have to think not only of providing proficient mathematicians, but also of using mathematical training for the development of men, for the cultivation of exactness of thought, and the power of grasping situations and dealing with problems that arise in life. But the furtherance of mathematical thought is no longer secondary. Perhaps in times past we were too much occupied with exercises of skill and ingenuity, too much, some of us, engrossed with the struggle for absolute perfection in the mastery of limited curricula, too serenely contented with the older analysis. But our slowness in assimilating new ideas by adequate study of the writings of the great masters of the newer analysis is over. Here as elsewhere a younger mathematician now realises that, after grounding himself in common knowledge, he must choose his department of higher study, must acquaint himself by prolonged effort with original authorities, and then produce for himself. A Congress like this will, I feel sure, greatly assist those who are inculcating the sound doctrine.

There is one special reason why I value the opportunity which has been given me of saying these few words of welcome. Like yourselves, ladies and gentlemen from abroad, I am a guest at Cambridge. Not being a Cambridge man, I can let myself say that, while Cambridge has had the honour of inviting you and has the pleasure of entertaining you, the welcome is extended to you by all the mathematicians of the United Kingdom. Our other universities, and my own of Oxford in particular, have Mathematical Faculties of which they are proud, and they do not sink their individualities in their consciousness of the greatness of that of Cambridge. But we owe many times more to Cambridge in the domain of mathematics than Cambridge owes to us. We come here for inspiration, and not infrequently for men. The small band of Oxonians here now will go back grateful to their Cambridge hosts, and yours, for one more benefit conferred in the opportunity of meeting you.

I will delay you no longer, ladies and gentlemen, except to refer in one word to the unspeakable loss sustained in the death of Henri Poincaré, whom in this Section we think of as the prince of analysts, and whom we had hoped to see here today. A higher power has ordered otherwise.

**2. Section II. Geometry.**

This Section met at 9.30. Dr Henry F Baker was in the Chair. Mr Arthur Lee Dixon was elected Secretary, and Dr Wilhelm Blaschke and Dr Enrico Bompiani were elected Assistant Secretaries. Professor Francesco Severi was elected Chairman for Saturday, 24 August.

The Chairman addressed the Section as follows:

Ladies and Gentlemen,

We are met here this morning to begin the work of the Section of Geometry. I believe it has been the custom that the President of the first meeting of a Section should be named by the Organising Committee, and I have been asked to undertake the duty. I beg you to permit me to make a few introductory remarks.

It will be the duty of the meeting, before we break up, to choose a President for the meeting of tomorrow of this Section of Geometry. It will also be necessary to elect a secretary for the Section of Geometry, and one or more assistant secretaries; the secretaries will hold their offices until the end of the Congress. Before I sit down I will suggest to you names for your consideration; and my remarks are to some extent directed to giving reasons for the name I intend to suggest for President tomorrow.

We in England have known geometers. Here Cayley lived and worked; and all of you know the name of Cayley, as you know the name of Salmon, who lived in Dublin, but was in close relation with Cayley. Today we have Sir Robert Ball, who has written a large book on the theory of linear complexes. I desire to express our regret that he is prevented by illness from being present. We have also many younger geometers, working in various directions. I am sure that these all join with me in saying how much honoured we in this country feel by the presence at the Congress of so many distinguished geometers from other lands. It is in order that we may express our gratification at their presence that I wish to say some words before the papers are communicated.

Of the recent progress of geometry in several directions you will hear from those who will present papers in this Section. There is, however, one side of geometry with which my own studies in the theory of algebraic functions have brought me into contact - the Theory of Algebraic Curves and Surfaces. I should like to say to you that I think extraordinary advances have recently been made in this regard, and to mention in a few words some of the more striking results. The history of the matter seems to me extremely interesting and, as has often happened before, the success has been achieved by the union of two streams of thought, which, though having a common origin, had for some time flowed apart.

It is a commonplace that the general theory of Higher Plane Curves, as we now understand it, would be impossible without the notion of the genus of a curve. The investigation by Abel of the number of independent integrals in terms of which his integral sums can be expressed may thus be held to be of paramount importance for the general theory. This was further emphasised by Riemann's consideration of the notion of birational transformation as a fundamental principle.

After this two streams of thought were to be seen.

First Clebsch remarked on the existence of an invariant for surfaces, analogous to the genus of a plane curve. This number he defined by a double integral; it was to be unaltered by birational transformation of the surface. Clebsch's idea was carried on and developed by Noether. But also Brill and Noether elaborated in a geometrical form the results for plane curves which had been obtained with transcendental considerations by Abel and Riemann. Then the geometers of Italy took up Noether's work with very remarkable genius, and carried it to a high pitch of perfection and clearness as a geometrical theory. In connection therewith there arose the important fact, which does not occur in Noether's papers, that it is necessary to consider a surface as possessing two genera; and the names of Cayley and Zeuthen should be referred to at this stage.

But at this time another stream was running in France. Picard was developing the theory of Riemann integrals - single integrals, not double integrals - upon a surface. How long and laborious was the task may be judged from the fact that the publication of Picard's book occupied ten years - and may even then have seemed to many to be an artificial and unproductive imitation of the theory of algebraic integrals for a curve. In the light of subsequent events, Picard's book appears likely to remain a permanent landmark in the history of geometry.

For now the two streams, the purely geometrical in Italy, the transcendental in France, have united. The results appear to me at least to be of the greatest importance.

Will you allow me to refer to some of the individual results - though with the time at my disposal I must give them roughly and without defining the technical terms?

Castelnuovo has shown that the deficiency of the characteristic series of a linear system of curves upon a surface cannot exceed the difference of the two genera of the surface, Enriques has completed this result by showing that for an algebraic system of curves the characteristic series is complete. Upon this result, and upon Picard's theory of integrals of the second kind, Severi has constructed a proof that the number of Picard integrals of the first kind upon a surface is equal to the difference of the genera. The names of Humbert and of Castelnuovo also arise here. Picard's theory of integrals of the third kind has given rise in Severi's hands to the expression of any curve lying on a surface linearly in terms of a finite number of fundamental curves. Enriques showed that the system of curves cut upon a plane by adjoint surfaces of order

These results, here stated so roughly, are, you see, of a very remarkable kind. They mean, I believe, that the theory of surfaces is beginning a vast new development. I have referred to them to emphasise the welcome which we in England wish to express to our distinguished foreign guests, whose presence here will, we believe, stimulate English geometry to a new activity. In particular we are very glad to have Professor Severi present with us in this room this morning. I will venture to propose to you that he be asked to act as Chairman of this Section of Geometry tomorrow, Saturday.

We are met here this morning to begin the work of the Section of Geometry. I believe it has been the custom that the President of the first meeting of a Section should be named by the Organising Committee, and I have been asked to undertake the duty. I beg you to permit me to make a few introductory remarks.

It will be the duty of the meeting, before we break up, to choose a President for the meeting of tomorrow of this Section of Geometry. It will also be necessary to elect a secretary for the Section of Geometry, and one or more assistant secretaries; the secretaries will hold their offices until the end of the Congress. Before I sit down I will suggest to you names for your consideration; and my remarks are to some extent directed to giving reasons for the name I intend to suggest for President tomorrow.

We in England have known geometers. Here Cayley lived and worked; and all of you know the name of Cayley, as you know the name of Salmon, who lived in Dublin, but was in close relation with Cayley. Today we have Sir Robert Ball, who has written a large book on the theory of linear complexes. I desire to express our regret that he is prevented by illness from being present. We have also many younger geometers, working in various directions. I am sure that these all join with me in saying how much honoured we in this country feel by the presence at the Congress of so many distinguished geometers from other lands. It is in order that we may express our gratification at their presence that I wish to say some words before the papers are communicated.

Of the recent progress of geometry in several directions you will hear from those who will present papers in this Section. There is, however, one side of geometry with which my own studies in the theory of algebraic functions have brought me into contact - the Theory of Algebraic Curves and Surfaces. I should like to say to you that I think extraordinary advances have recently been made in this regard, and to mention in a few words some of the more striking results. The history of the matter seems to me extremely interesting and, as has often happened before, the success has been achieved by the union of two streams of thought, which, though having a common origin, had for some time flowed apart.

It is a commonplace that the general theory of Higher Plane Curves, as we now understand it, would be impossible without the notion of the genus of a curve. The investigation by Abel of the number of independent integrals in terms of which his integral sums can be expressed may thus be held to be of paramount importance for the general theory. This was further emphasised by Riemann's consideration of the notion of birational transformation as a fundamental principle.

After this two streams of thought were to be seen.

First Clebsch remarked on the existence of an invariant for surfaces, analogous to the genus of a plane curve. This number he defined by a double integral; it was to be unaltered by birational transformation of the surface. Clebsch's idea was carried on and developed by Noether. But also Brill and Noether elaborated in a geometrical form the results for plane curves which had been obtained with transcendental considerations by Abel and Riemann. Then the geometers of Italy took up Noether's work with very remarkable genius, and carried it to a high pitch of perfection and clearness as a geometrical theory. In connection therewith there arose the important fact, which does not occur in Noether's papers, that it is necessary to consider a surface as possessing two genera; and the names of Cayley and Zeuthen should be referred to at this stage.

But at this time another stream was running in France. Picard was developing the theory of Riemann integrals - single integrals, not double integrals - upon a surface. How long and laborious was the task may be judged from the fact that the publication of Picard's book occupied ten years - and may even then have seemed to many to be an artificial and unproductive imitation of the theory of algebraic integrals for a curve. In the light of subsequent events, Picard's book appears likely to remain a permanent landmark in the history of geometry.

For now the two streams, the purely geometrical in Italy, the transcendental in France, have united. The results appear to me at least to be of the greatest importance.

Will you allow me to refer to some of the individual results - though with the time at my disposal I must give them roughly and without defining the technical terms?

Castelnuovo has shown that the deficiency of the characteristic series of a linear system of curves upon a surface cannot exceed the difference of the two genera of the surface, Enriques has completed this result by showing that for an algebraic system of curves the characteristic series is complete. Upon this result, and upon Picard's theory of integrals of the second kind, Severi has constructed a proof that the number of Picard integrals of the first kind upon a surface is equal to the difference of the genera. The names of Humbert and of Castelnuovo also arise here. Picard's theory of integrals of the third kind has given rise in Severi's hands to the expression of any curve lying on a surface linearly in terms of a finite number of fundamental curves. Enriques showed that the system of curves cut upon a plane by adjoint surfaces of order

*n*- 3, when*n*is the order of the fundamental surface, if not complete, has a deficiency not exceeding the difference of the genera of the surface. Severi has given a geometrical proof that this deficiency is equal to the difference of the genera, a result previously deduced by Picard, with transcendental considerations, from the assumption of the number of Picard integrals of the first kind. The whole theory originally arose, as has been said, with Clebsch's remark of a numerical invariant of birational transformation; conversely it is a matter of the profoundest geometrical interest to state in terms of invariants the sufficient conditions for the birational transformation of one surface into another. I might make reference to the Zeuthen-Segre invariant, which has been extended by Castelnuovo to the case of an algebraic system of curves. I will allow myself to mention, out of a vast number of results, one striking theorem. Enriques and Castelnuovo have shown that a surface which possesses a system of curves for which what may be called the canonical number, 2π - 2 -*n*, where π is the genus of a curve and*n*the number of intersections of two curves of the system, is negative, can be transformed birationally to a ruled surface. There is one other result I will refer to. On the analogy of the case of plane curves, and of surfaces in three dimensions, it appears very natural to conclude that if a rational relation, connecting, say,*m*+ 1 variables, can be resolved by substituting for the variables rational functions of m others, then these m others can be so chosen as to be rational functions of the*m*+ 1 original variables. Enriques has recently given a case, with*m*= 3, for which this is not so.These results, here stated so roughly, are, you see, of a very remarkable kind. They mean, I believe, that the theory of surfaces is beginning a vast new development. I have referred to them to emphasise the welcome which we in England wish to express to our distinguished foreign guests, whose presence here will, we believe, stimulate English geometry to a new activity. In particular we are very glad to have Professor Severi present with us in this room this morning. I will venture to propose to you that he be asked to act as Chairman of this Section of Geometry tomorrow, Saturday.

**3. Section III (a). Mechanics, Physical Mathematics, Astronomy.**

The subsection met at 9.30. Professor H Lamb was in the Chair. Mr F J M Stratton was elected Secretary, and Professor Giulio Andreoli and Dr Ludwig Föppl were elected Assistant Secretaries. Prince B Galitzin was elected Chairman for Saturday, 24 August.

The Chairman addressed the subsection as follows:

You will not expect that I should say more than a few words before we enter on the business of the meeting. But as I have the privilege of being, like most of those present, a visitor and a guest, I may be allowed, or rather you will wish me, to give expression to a thought which is doubtless present to the minds of those who have followed the migrations of the Congress from place to place. The University of Cambridge has a long and glorious record in connection with our special subject of Mechanics. I need not repeat the words which fell from our President yesterday; but it must be a matter of peculiar interest and satisfaction to the members of our Section that they are at length assembled in a place so intimately associated with the names of illustrious leaders in our science.

One other point I would ask leave to touch upon. In spite of the process of subdivision which has been carried out, the field covered by our Section is still a very wide one. It has been said that there are two distinct classes of applied mathematicians; viz. those whose interest lies mainly in the purely mathematical aspect of the problems suggested by experience, and those to whom on the other hand analysis is only a means to an end, the interpretation and coordination of the phenomena of the world. May I suggest that there is at least one other and an intermediate class, of which the Cambridge school has furnished many examples, who find a kind of aesthetic interest in the reciprocal play of theory and experience, who delight to see the results of analysis verified in the flash of ripples over a pool, as well as in the stately evolutions of the planetary bodies, and who find a satisfaction, again, in the continual improvement and refinement of the analytical methods which physical problems have suggested and evoked? All these classes are represented in force here today; and we trust that by mutual intercourse, and by the discussions in this Section, this Congress may contribute something to the advancement of that Science of Mechanics, in its widest sense, which we all have at heart.

One other point I would ask leave to touch upon. In spite of the process of subdivision which has been carried out, the field covered by our Section is still a very wide one. It has been said that there are two distinct classes of applied mathematicians; viz. those whose interest lies mainly in the purely mathematical aspect of the problems suggested by experience, and those to whom on the other hand analysis is only a means to an end, the interpretation and coordination of the phenomena of the world. May I suggest that there is at least one other and an intermediate class, of which the Cambridge school has furnished many examples, who find a kind of aesthetic interest in the reciprocal play of theory and experience, who delight to see the results of analysis verified in the flash of ripples over a pool, as well as in the stately evolutions of the planetary bodies, and who find a satisfaction, again, in the continual improvement and refinement of the analytical methods which physical problems have suggested and evoked? All these classes are represented in force here today; and we trust that by mutual intercourse, and by the discussions in this Section, this Congress may contribute something to the advancement of that Science of Mechanics, in its widest sense, which we all have at heart.

**4. Section III (b). Economics, Actuarial Science, Statistics.**

The subsection met at 9.30. Professor Francis Ysidro Edgeworth was in the Chair. Professor Arthur Lyon Bowley was elected Secretary. Dr William Fleetwood Sheppard was elected Chairman for Saturday, 24 August.

The Chairman addressed the subsection as follows:

The first duty of the Chairman is, on behalf of English members of the Congress, to welcome visitors from foreign countries. There is a particular propriety in the expression of such a welcome by our subsection. For we are particularly benefited by the presence of visitors from distant countries. The advantage which is obtained by exchange of ideas with original minds educated in different ways is at a maximum when the subjects dealt with, like most of ours, are somewhat dialectical and speculative. Speaking of visitors, I cannot forget that many of the English members of the Congress, like myself, are visitors to Cambridge. Visitors of all. nationalities will be unanimous in expressing their appreciation of Cambridge hospitality. Towards our subsection this hospitality has a peculiar delicacy. It is not merely that we are admitted to the privileges of Hall and Commonroom. We have received an invitation even more grateful to some of us. There are some classes of us who have hitherto, so to speak, "sat below the salt" at the feast of reason. Economic Science is now in the position of that humble but deserving one who was invited to " go up higher." It is a proud day for Mathematical Economics, the day on which it enters the Congress of Mathematicians

As for the Actuarial branch of our subsection the Science founded by Halley, like the sun, does not require recognition. Still even in the midst of rigid actuarial formulae there is an element of the calculus of probabilities. Thus even with respect to this department what may be described as the more human side of mathematics is now recognised. The Calculus of Probabilities has been described by purists of old as the opprobrium of mathematics. It will be the part of this subsection, with the valuable cooperation of foreign statisticians, to dispel this prejudice.

*pari passu*with other mathematical subjects. There is a propriety in this recognition being made at Cambridge, in which University Dr Marshall has shown that mathematical reasoning in Economics may be not only brilliant but fruitful. The cultivators of our second branch, Mathematical Statistics, have also reason to be satisfied with their reception. Cambridge has indeed lately shown her appreciation of their work in a very solid fashion by establishing a lectureship in Statistics. I trust that the designated lecturer will take part in our debates.As for the Actuarial branch of our subsection the Science founded by Halley, like the sun, does not require recognition. Still even in the midst of rigid actuarial formulae there is an element of the calculus of probabilities. Thus even with respect to this department what may be described as the more human side of mathematics is now recognised. The Calculus of Probabilities has been described by purists of old as the opprobrium of mathematics. It will be the part of this subsection, with the valuable cooperation of foreign statisticians, to dispel this prejudice.

**5. Section IV (a). Philosophy and History.**

The subsection met at 9.30. The Hon Bertrand Arthur William Russell was in the Chair. Professor Edward Vermilye Huntington and Professor Maurice Fréchet were elected Secretaries. Professor August Gutzmer was elected Chairman for Saturday, 24 August.

The Chairman addressed the subsection as follows:

Ladies and Gentlemen,

In opening the meetings of this Section, I desire to say one word of welcome to the distinguished visitors whom we are glad to see amongst us. The philosophy of mathematics has made extraordinarily rapid advances in recent times, and I am happy to see that many of those to whom these advances owe most are taking part in our meetings. Some unavoidable absences are to be deplored; among these, the illustrious name of Georg Cantor will occur to all. I had hoped, but in vain, that we might have been honoured by the presence of Frege, who, after many years of indomitable perseverance, is now beginning to receive the recognition which is his due. In common with other sections, we cannot but feel how great a loss we have sustained by the death of Henri Poincaré, whose comprehensive knowledge, trenchant wit, and almost miraculous lucidity gave to his writings on mathematical philosophy certain great qualities hardly to be found elsewhere. The work of the pioneers has been great, not only through its actual achievement, but through the promise of an exact method and a security of progress of which, I am convinced, the papers and discussions which we are to hear will afford renewed evidence.

In opening the meetings of this Section, I desire to say one word of welcome to the distinguished visitors whom we are glad to see amongst us. The philosophy of mathematics has made extraordinarily rapid advances in recent times, and I am happy to see that many of those to whom these advances owe most are taking part in our meetings. Some unavoidable absences are to be deplored; among these, the illustrious name of Georg Cantor will occur to all. I had hoped, but in vain, that we might have been honoured by the presence of Frege, who, after many years of indomitable perseverance, is now beginning to receive the recognition which is his due. In common with other sections, we cannot but feel how great a loss we have sustained by the death of Henri Poincaré, whose comprehensive knowledge, trenchant wit, and almost miraculous lucidity gave to his writings on mathematical philosophy certain great qualities hardly to be found elsewhere. The work of the pioneers has been great, not only through its actual achievement, but through the promise of an exact method and a security of progress of which, I am convinced, the papers and discussions which we are to hear will afford renewed evidence.

**6. Section IV (b). Didactics.**

The subsection met at 9.30. Mr Charles Godfrey was in the Chair. Professor George Alexander Gibson was elected Secretary, and Messrs J Franklin and E A Price were elected Assistant Secretaries. Professor A Gutzmer was elected Chairman for the joint meeting of Sections IV (a) and IV (b), and Professor Emanuel Czuber for the subsequent meeting.

The Chairman addressed the subsection as follows:

Gentlemen,

This is the opening meeting of Subsection IV (b), the subsection engaged in the discussion of didactical questions. The subsection will hold five meetings. Three of these meetings will be taken up by proceedings arising out of the activities of the International Commission on Mathematical Teaching - namely the meetings of this morning, of Monday afternoon, and of Tuesday morning. Two meetings remain, those of Saturday morning and Monday morning. On Saturday morning this subsection will join with the subsection for philosophical questions to discuss the very important question - how far it is expedient to introduce into school teaching the consideration of the fundamentals of mathematics.

After the words of welcome spoken by Sir George Darwin yesterday, no further words of mine should be needed to make our visitors from abroad feel that they are 'at home' among us. But it is fitting that I should avail myself of this occasion to offer to our visitors a very special welcome on behalf of the Mathematical teachers of this country. We Mathematical teachers welcome you, first because we are glad to have you with us and because we are glad to have the opportunity of making new friendships. We welcome you for another reason - because there is much that we can learn from you in the exercise of our craft. M Bourlet has expressed the opinion that it is futile to transplant the teaching methods of one country into another, and to expect that these methods will always flourish in a new environment. I agree with his remarks; but I repeat that we have much to learn from you, and I assure you that many of us propose so to learn.

It is a matter of deep regret to all of us that our natural leader, Professor Klein, is unable to be present at this Congress. I will not anticipate the resolution of regret that Sir George Darwin will submit to you. For myself, I have done my best to acquaint myself with Professor Klein's views on Mathematical teaching, with which I am strongly in sympathy. If I may try to characterise in mathematical language the leading motif of the movement of which Professor Klein is the leader, it is this - that mathematical teaching is a function of two variables: the one variable is the subject-matter of mathematics, the other variable is the boy or girl to whom the teaching is addressed; the neglect of this second variable is at the root of most of the errors that Professor Klein combats.

I learn from a letter addressed to Sir George Darwin that there is one matter which interests Professor Klein greatly and that he would have desired to call the attention of the subsection to it. It is the publication of the Encyclopaedic work

The meeting will now be asked to receive the report of the International Commission, and I hope that I shall be allowed to delegate my duties as Chairman to Professor D E Smith, to whose initiative the creation of the International Commission is due.

This is the opening meeting of Subsection IV (b), the subsection engaged in the discussion of didactical questions. The subsection will hold five meetings. Three of these meetings will be taken up by proceedings arising out of the activities of the International Commission on Mathematical Teaching - namely the meetings of this morning, of Monday afternoon, and of Tuesday morning. Two meetings remain, those of Saturday morning and Monday morning. On Saturday morning this subsection will join with the subsection for philosophical questions to discuss the very important question - how far it is expedient to introduce into school teaching the consideration of the fundamentals of mathematics.

After the words of welcome spoken by Sir George Darwin yesterday, no further words of mine should be needed to make our visitors from abroad feel that they are 'at home' among us. But it is fitting that I should avail myself of this occasion to offer to our visitors a very special welcome on behalf of the Mathematical teachers of this country. We Mathematical teachers welcome you, first because we are glad to have you with us and because we are glad to have the opportunity of making new friendships. We welcome you for another reason - because there is much that we can learn from you in the exercise of our craft. M Bourlet has expressed the opinion that it is futile to transplant the teaching methods of one country into another, and to expect that these methods will always flourish in a new environment. I agree with his remarks; but I repeat that we have much to learn from you, and I assure you that many of us propose so to learn.

It is a matter of deep regret to all of us that our natural leader, Professor Klein, is unable to be present at this Congress. I will not anticipate the resolution of regret that Sir George Darwin will submit to you. For myself, I have done my best to acquaint myself with Professor Klein's views on Mathematical teaching, with which I am strongly in sympathy. If I may try to characterise in mathematical language the leading motif of the movement of which Professor Klein is the leader, it is this - that mathematical teaching is a function of two variables: the one variable is the subject-matter of mathematics, the other variable is the boy or girl to whom the teaching is addressed; the neglect of this second variable is at the root of most of the errors that Professor Klein combats.

I learn from a letter addressed to Sir George Darwin that there is one matter which interests Professor Klein greatly and that he would have desired to call the attention of the subsection to it. It is the publication of the Encyclopaedic work

*Die Kultur der Gegenwart*which is in course of compilation under his direction. This work will consist of a series of volumes in which every branch of culture is explained by experts in non-technical language, so that the articles will be within the reach of the reader of general education. This undertaking does not, it is true, appertain to education in the narrower sense of the word, but it does not seem too great an extension of the word to regard it as belonging to our special division. Dr Klein remarks in his letter that it was a matter of much difficulty to determine how so specialised a subject as mathematics could be made a suitable one for memoirs of the general character described, but he is glad to say that a good beginning has been made by Dr Zeuthen of Copenhagen in an article on the Mathematics of Classical Times and of the Middle Ages. Those who are interested in this will be able to see copies of the article in the Exhibition.The meeting will now be asked to receive the report of the International Commission, and I hope that I shall be allowed to delegate my duties as Chairman to Professor D E Smith, to whose initiative the creation of the International Commission is due.