1928 ICM - Bologna

1928 International Congress of Mathematicians - Bologna, Italy

The International Congress of Mathematicians was held in Bologna, Italy from 3 September to 10 September 1928. The Congress was attended by 836 full members, 280 family members, giving a total of 1116. We give below a version of:
  1. Preparations for the Congress
  2. Programme of the Congress
  3. Opening events of the Congress
  4. Continuation of the Congress
  5. Closing Session of the Congress
Before presenting the material, we give a short Preface.
Preface by EFR and JJOC.

The main difficulty with this Congress was the problem over the exclusion of 'ex-enemy' countries. The Organising Committee was put in an almost impossible position and did a remarkable job walking a tightrope between those wanting to continue the exclusions and those desperate to make it truly international again. The Organising Committee sets out in detail the extraordinary difficulties it faced - see below. Notice that on the title page of the Proceedings appears (VI). This Congress considered itself the Sixth International Congress of Mathematicians, meaning that it did not regard the 1920 and 1924 congresses to be International Congresses because they were not open to all. Up to 1916 the Congresses had been numbered 1st, 2nd, 3rd, 4th, 5th. The Proceedings of 1920 and 1924 have no number, 1928 has (VI), and no later Congress numbers itself to avoid the difficulty. The Fascist government of Italy gave the Congress strong financial support looking to impress the international visitors with their political system.

1.       Preparation for the Congress.

1.1. History.

The series of International Congresses of Mathematics, which began in Zurich in 1897, interrupted during the war, was resumed and the Congresses of Strasbourg of 1920 and Toronto of 1924, called by the International Mathematical Union (issued by the International Research Council), with the exclusion of the scientists of German, Bulgarian, Austrian and Hungarian nationality.

In the session of 15 August 1924, held in Toronto by the delegates of the International Mathematical Union, the following motion was presented by the delegates of the United States of America:
The American Section of the International Mathematical Union request the International Research Council to consider whether the time is ripe for the removal of restrictions on membership now imposed by the rules of the Council.
That motion was supported by Italy, Denmark, the Netherlands, Sweden, Norway and Great Britain.

In that same session, professor Salvatore Pincherle was elected president of the International Mathematical Union and president of the Italian Mathematical Union, and it was decided that the location of the future Congress, which was to take place in 1928, was to be set by decision of the Board of the International Mathematical Union by the year 1926.

The choice fell on the city of Bologna, and this designation was communicated by the Secretary General of the International Mathematical Union in a circular dated November 1926. Meanwhile the International Research Council, in the session of 29 June 1926, had decided to lift any restrictions on admission, among the unions belonging to it, of the hitherto non-acceding States, and had invited Germany, Austria, Bulgaria and Hungary to participate in the Council.

The presidency of the Italian Mathematical Union, which had the serious task of preparing the Congress, therefore decided to resume the traditions of the pre-war International Congresses, by removing any exclusion dependent on political reasons, placed the future Congress under the auspices of the University of Bologna, and entrusted its organisation to a local committee, made up of professors from the university itself and leading personalities of the city.

This Committee was established in the session of 18 January 1927, appointing the Rector pro tempore of the University of Bologna as President, Professor Ettore Bortolotti as Secretary, and G Borsari, the secretary of the University of Bologna at treasurer; he appointed an Executive Commission within it, headed by Professor Pincherle, President of the Italian Mathematical Union and of the International Mathematical Union and decided to request the high patronage of His Majesty the King of Italy and the honorary presidency of His Excellency the Head of the Government.

Both of these requests were graciously and favourably received.

The Executive Commission recognised the following points as essential:
1. - Establish relationships of cordial connection with all the main associations, academies, and scientific institutions of each country, to obtain membership, collaboration, support, and to ensure the participation of the most illustrious scientists in the Congress.

2. - Organise scientific work.

3. - Prepare a dignified and hospitable welcome for members of the Congress and their families.

4. - Provide the funds necessary for the preparation and conduct of the Congress and for the printing of the Proceedings.
1.2. Relations with scientific institutes.

As regards the first point, the Executive Commission first issued a preliminary notification, to announce the Congress, the place, the date, the sections and the topics. This notification, written in five languages and sent, together with appropriate accompanying letters, to the presidents and secretaries of all the scientific institutions of all the countries where the mathematical sciences are cultivated, was greatly welcomed everywhere. Following the example of the American Mathematical Society, all the societies and academies contacted offered to distribute to their members both the notification announcement and any further communications, and to publish news about the Congress in their Proceedings.

Of each circular, about ten thousand copies were distributed, and it can be said that, in every part of the world where a mathematical school existed, news of the Congress of Bologna arrived.

The largest German mathematical society, Deutsche mathematische Vereinigung, also distributed to its partners the "preliminary notification", which is contained in an issue of its Jahresberichte.

However, a strong opposition soon arose from a few, but authoritative representatives of German science, who, reviving sad episodes of the immediate post-war period, believed they saw, in the reports of the International Mathematical Union (from which the Congress had been called) with the International Research Council, a political dependence on opposing nationalist groups, to which they attributed the party which had boycotted German science.

These unfortunate protests made the field noisy, not only in the countries of German nationality, but also in those that were neutral in the war, and their echo extended to England and America, causing a not insignificant danger to the success of the Congress; since from Holland, Denmark, Sweden, from the most authorised groups of England and the United States, had made it known, in a formal way, to the presidency of the Executive Commission, that political exclusions would no longer be tolerated, and that a Congress that had not been international, in the most absolute sense of the word, would have met with general abstention.

Overcoming this ill-founded mistrust was not easy or quick. In addition to the private talks with some of the most authoritative scientists who visited our city, the large and uninterrupted correspondence by letter, the circulars that announced in their most significant details the conduct of the future Congress, and the invitations that, without distinction of nationality, benefited from this, notices issued by the Rector of the University of Bologna, an extraneous personality and above any political influence, were made to academies and scientific societies around the world: finally, the chosen host of foreign scientists cordially accepted the invitation to deal general interest issues in the combined section lectures.

When the storm seemed to have passed, a more formidable threat came from the opposite side.

The regulatory provisions which had been established for the International Mathematical Union in the immediate post-war period did not authorise anyone to invite people to International Mathematical Congresses other than: "scientific groups from countries belonging to the International Research Council". These provisions, which had already been overcome in the Toronto Congress, in which, without any opposition, scientific groupings had participated, and official delegations had been sent by countries not belonging to the International Council, such as Russia, Spain, India and Georgia, on the other hand, were recalled, and in the most peremptory tone, when it became known that the invitation made by the President of the Committee organising the future Congress of Bologna, had also been extended to Germany, a country that had not yet responded to the invitation to join the International Research Council.
This serious breach," the Secretary General of the International Mathematical Union wrote in the letter of 29 May 1928 to the President of the Executive Commission, "makes all these invitations illegal. ... Under the conditions in which these invitations were made, it can no longer be said that the Bologna Congress is a Congress belonging to the International Mathematical Union. Consequently, and after having conferred further with the President of the International Research Council, it is impossible for me to invite our members to the Congress that the University of Bologna will have organised in this city under the chairmanship of its Rector. In addition, to enlighten our members and let them know the true state of affairs, I will send them a copy of your letter of 26 April and this response.
In response to this protest, the President of the Executive Commission (also President of the International Mathematical Union) addressed the President of the International Research Council (Professor Émile Picard of Paris) a letter which exposed in the most explicit way, and at the same time justified the line of conduct of the Congress organisers. But the response of the President of the International Research Council confirmed, albeit with some attenuation of form, the statements of the Secretary General.

For the letter sent by the President of the International Mathematical Union, Salvatore Pincherle, to the President of the International Research Council, Émile Picard, see THIS LINK.

These quarrels did not in the least divert the preparation of the Congress from the predetermined line of conduct, in which, moreover, before then, neither the International Mathematical Union, nor the International Research Council had in any way participated. Under the auspices of the University of Bologna, the Organising Committee persevered in its work, intended to reconcile minds, to bring together the scientists of the countries that the war had divided, and to re-establish those cordial relationships of connection, traditional among mathematicians in the Congresses held before the war.

Nor was the greater concern of the disclosure of a circular letter, published a few days before the opening of the Congress by a distinguished scientist, but a stubborn opponent (of non-German nationality) to alienate the supporters of the German side from the Congress; a letter recalling the period in which, in a time that had now passed, ostracism was banished by the International Congresses of German science. The letter concluded
In view of these words, each mathematician should consider to what extent participation in the planned Congress is possible without mocking the memory of Gauss and Riemann, of the cultural character of mathematical science, and the independence of the human spirit.
But despite these oppositions, the efforts of the Organising Committee to achieve the goal were fully successful, to the point of exceeding their expectations.

The work done in opposing it did not prevent the German nation from having the most numerous representation (after the Italian one) in Bologna among those attending the Congress; in the same way that the interdiction of the International Research Council and the General Secretariat of the International Mathematical Union could not prevent 15 of the 19 nations belonging to the International Research Council being represented at the Congress and that 209 of the Scientific Institutions and 22 of the countries invited on behalf of the Rector of the University of Bologna, were represented by officially appointed delegates; and finally that more than 1100 members of congress agreed to Bologna from all over the world! But the episodes just mentioned give an idea of the difficulties that the organisation of the Congress had to face in order to overcome the crisis of passage from the restricted regime of nationalistic exclusions, which the war had left as a legacy to the Mathematical Congresses, to a regime of full independence from every political regime, as befits a truly international scientific congress.

And having overcome that crisis is one of the major successes of the Bologna Congress. This has frankly recognised authoritative men from opposing parties. Among the many letters received, it seems appropriate to report that of one of the most representative scientists on the German side, a not indifferent testimony of the work of the Bologna Committee in all phases of the laborious preparation, which with sincere frankness, in the very act in which it announced that his official capacity forbade him to speak at the Congress, he wrote: "... So you will still have the feeling of fame for having taken the first and a very big step forward in the recovery of the circumstances. Thanks to your work, the next congress will no longer need any thanks. Should the full success of your effort only mature in four years, you will have the fame of the brave pioneer who, despite all the difficulties for three years, managed to follow the path once taken wisely and energetically, to be bravely and indomitably."

And at the end of the Congress, a congressman, distinguished mathematician and Rector of an important German University, wrote to the President:
... you will receive warm thanks from all sides that your tactful behaviour has managed to completely leave politics out of the game and that mathematicians of all countries unite for purely scientific endeavours.
1.3. Organisation of scientific work.

The Executive Commission established that the scientific work of the Congress was accomplished by means of lectures made in plenary sessions by recognised scientists, expressly invited by the President of the Organising Committee upon indication of the Executive Commission, on topics of general interest, and Section Communications to which all mathematics science students were admitted freely, without distinction of nationality and school.

In order to organise the Section Communications, the Executive Commission appointed for each Section several Introducers, who, taking advantage of their personal knowledge and the authority of their name in the scientific field each of them particularly cultivated, could assure Congress of the intervention of the most illustrious scientists and experts of greater following.

The Communications offered were in exceptional numbers out of the ordinary (including the one received by the Secretariat after the publication of the Diary, not less than 419). The Congress Secretariat invited all those who wished to present Communications, to make known, together with the title, a clear and brief presentation of the topic; and, in possession of the arguments of all the Communications presented, proceeded to a further subdivision of the Sections that had initially been established, and distributed the Communications so that in the same session of each Section, related or related topics were dealt with between them, and that there was no great disparity in the number of Communications assigned to each individual session.

This resulted in the following groupings:
SECTION I. Analysis (four subsections): I-A. Number theory. - Algebras, Matrices. - Discontinuous groups. - Algebraic equations. - Algebraic functions. - I-B. Functions of a real variable. - Set theory. Topological considerations. - Generalised integrals. - Summation of series and integrals. - Functions and almost periodic sequences. - I-C. Differential equations, difference equations, integral equations, integro-differential equations. - Calculus of variations. - Functional analysis. - I-D. Analytical functions. - Expansion in series. - Conformal representations. - Topology.

SECTION II. Geometry (two subsections): II-A. Topological issues related to algebraic geometry. - Groups of Cremonian transformations, birational, contact. .... - General research on algebraic curves and surfaces. - Special researches. - II-B. Topology, in relation to differential geometry. - Riemann geometry and its extensions. - Projective-differential geometry. - Geometry of the sphere and the straight line. - Various.

SECTION III. Mechanics (two subsections): III-A. Celestial mechanics. Astronomy. - Mechanics of systems. - Relativistic mechanics. - Electrical engineering. - III-B. Hydrodynamics. - Elasticity. - Equations of mathematical physics. - Various.

SECTION IV. Actuarial mathematics (two subsections): IV-A. Probability calculus. - Mathematical Statistics. - Error theory. - Averages and interpolations. - IV-B. Mathematical economics and actuarial science.

SECTION V. Engineering: Hydraulics. - Aerodynamics. - Buildings (Bridges). - Cartography. - Industrial applications.

SECTION VI. Elementary mathematics: Mathematical logic. - Didactic issues, International Commission for Mathematical Teaching. - Pedagogy and mathematical methodology. - Various.

SECTION VII. History of Mathematics. Philosophy: Mathematical philosophy. - History of mathematics in antiquity and in the Middle Ages. - History of mathematics in the Renaissance and in the modern era. - Mathematical bibliography.
In Section II-A, a group of reports on the progress of the various theories that make up Algebraic Geometry, promoted and coordinated by Professor Severi, presented special interest.

These reports were collected in the same session and are included in the Proceedings.

The subjects of the Communications were promptly announced collected in a volume and distributed to the members of Congress together with the Diary of the sessions, before the beginning of the Congress.

The dispositions for the ordering and the carrying out of the works were given with the circulars issued in November 1927 and in January 1928. Together, they constitute the following:

1.4. Congress Regulations.

All lovers of pure and applied mathematical sciences are invited to the International Congress of Mathematicians, to be held in Bologna from 3 to 10 September 1928 under the auspices of the University of Bologna.

Members of Congress. - The members of Congress are, either actual members, or associates. Full members who pay a registration fee of 50 Italian Lire have the right to make communications or proposals during Congress sessions, to take part in discussions and voting, to participate in ceremonies, celebrations, excursions, visits; will enjoy discounts on the price of travel on the Italian railways and shipping lines, reductions in hotels and restaurants; upon presentation of the conference card, they will be granted free of charge the visa of the RR Consular Authorities on passports, as well as a residence permit. They will have free admission to museums and galleries, and will be given free volumes of the Congress Proceedings.

Persons belonging to the family of a full member can participate in the Congress as associated members, by paying a registration fee of 25 Italian Lire. Associate members will not have free congress volumes; they will not be able to participate in the votes in the Congress, make communications or take part in the discussions; but they will be able to attend the sessions, take part in receptions, excursions, visits, etc., and will enjoy the discounts and all the other concessions and facilities given to the full members.

1.5. Meetings and Works of the Congress.

The solemn opening session will take place in the Aula magna of the ancient Archiginnasio in Bologna, on 3 September. The closing session will take place in Florence in the Salone dei Cinquecento in Palazzo Vecchio.

At the closing session, the General Assembly of full members will designate the venue for the future Congress.

All other sessions and meetings will be held in the University building and in the scientific institutes adjacent to it.

The Congress will take place with plenary sessions and Section meetings. In plenary sessions, lectures of general interest will be read by authoritative scholars of the various branches of science; Section meetings will read Communications on matters pertaining to the Section title.

The works will be distributed in 7 sections, which will be: 1) Arithmetic, Algebra, Analysis; 2) Geometry; 3) Mechanics, Astronomy, Geodesy, Geophysics, Mathematical physics, Theoretical physics; 4) Statistics, Mathematical Economics, Probability Calculus, Actuarial Science; 5) Engineering and Industrial applications; 6) Elementary Mathematics, Didactic Issues, Mathematical Logic; 7) Philosophy, History of mathematics.

It is understood that, by requiring the number of Communications, each Section may be divided into Subsections.

2.       Programme of the Congress.

Sunday 2 September.

21.30 - Introductory meeting of members of the Congress with the Members of the Italian Mathematical Union, in the halls of the Culture Club, kindly granted (via Mazzini, 45).

Monday 3 September.

10.00 - Solemn inaugural session in the Archiginnasio Aula Magna.

15.00 - First session - University of Bologna - Appointment of the President and Vice-Presidents.

16.00 - Lectures: D Hilbert - Probleme der Grundlegung der Mathematik.
- J Hadamard - Le développement et le rôle scientifique du Calcul fonctionnel. - U Puppini - Le bonifiche in Italia.

Tuesday 4th September.

9.00 - Lectures: E Borel - Le calcul des probabilités et les sciences exactes. - O Veblen - Differential Invariants and Geometry. - G Castelnuovo - La Geometria algebrica e la Scuola Italiana.

16.00 - Sections meeting.

Wednesday 5 September.

9.00 - Lectures: W H Young - The mathematical method and its limitations. - V Volterra - La teoria dei funzionali applicata ai fenomeni ereditari. - H Weyl - Darstellung kontinuirlichen Gruppen.

16.00 - Sections meeting.

21.30 - Reception offered by the Podestà.

Thursday 6 September.

9.00 - Lectures: V Kármán - Mathematische Probleme der modernen Aerodynamik. - L Tonelli - Contributo italiano alla Teoria delle funzioni di variabili reali. - L Amoroso - Le equazioni differenziali della dinamica economica.

16.00 - Sections meeting.

21.00 - Orchestral concert of historical Italian music.

Friday 7 September.

Visits to Ravenna - to Ferrara - to Lake Ledro.

Saturday 8 September.

9.00 - Lectures: M Fréchet - L'analyse generale et les espaces abstraits. - R Marcolongo - Leonardo da Vinci nella storia della matematica e della meccanica. - N Luzin - Sur les voies de la théorie des ensembles.

12.00 - Lunch at the Littoriale, offered by the Congress Organising Committee.

16.00 - Sections meeting.

22.00 - Reception offered by the National Government.

Sunday 9 September.

9.00 - Lecture: F Enriques - Continuità e discontinuità nella Geometria algebrica.

16.00 - Visit to the monuments of the city - Discovering a plaque in the paternal house of Scipione Dal Ferro, and a plaque in the church where Bonaventura Cavalieri was prior.

Monday 10 September.

6.25 - Departure for Florence.

11.00 - Closing session of the Congress in the Palazzo Vecchio in Florence. Lecture: G Birkhoff - Quelques éléments mathématiques de l'art.

3.       Opening events of the Congress.

Sunday 2 September.

At 21.00, in the premises of the Circolo di Coltura, kindly granted, the members of the Congress who had already arrived in Bologna were festively welcomed by the members of the Italian Mathematical Union. The mutual presentations were simple and cordial; the lively meeting continued until past midnight.

Monday 3 September.

On Monday 3 September 1928, at 10.00, in the Aula Magna of the ancient Archiginnasio, in the presence of His Royal Highness. the Duke of Bergamo representing His Majesty the King of Italy; of His Excellency the Hon Belluzzo, Minister of Public Instruction, representing the Head of the Government; of His Excellency the Prefect of the Province of Bologna; of His Excellency Cardinal Nasalli Rocca, Archbishop of Bologna, and of all local authorities, the International Congress of Mathematicians was solemnly inaugurated.

The Podestà of Bologna, Hon L Arpinati, offered to those present the greeting in the name of the city of Bologna.

Summary of the Podestà's speech:
Royal Highness, Eminence, Gentlemen.

It is with deep emotion that on behalf of the city that I have the honour of representing, I offer you best wishes. Bologna is grateful to you for having chosen it as the venue for your great conference; she is grateful for the act of recognition of the centuries-old glorious university tradition. Fascist Bologna is proud to welcome you and to be able to show you what it has become under the vivifying impulse of Fascism.

In your short stay in Italy, I hope you will be able to see and make an accurate account of our state and our spirit; and I hope that you, having finished the work of the Congress, returning home, will be able to keep a grateful memory of Bologna and Italy.
After the greeting of the Podestà, the Senator professor G Albini, Magnificent Rector of the University of Bologna and President of the Organising Committee read an oration in Latin which we omit.

Speech by Professor S Pincherle, President of the Executive Committee of the Congress:
Royal Highness, Excellencies, Ladies and Gentlemen,

An eloquent greeting has just been addressed, in the immortal language of Rome, to the many scholars of all countries who are gathered in this room. This greeting was pronounced by the Rector of the University of Bologna: and this is a fact which goes beyond a simple expression of courtesy, because for the intention of those who prepared this solemn meeting, this fact must represent the end of the state of unease, a state which was a consequence of the war and which has continued until now.

The International Congresses of Mathematicians, inaugurated in Zurich in 1897 and after which, Olympiads of thought, have succeeded each other for four years, were interrupted by the war. After the war, the International Mathematical Union wanted to renew the series; but this Union, influenced by a state of mind which the psychology of the aftermath of the war suffices to explain, if not to justify, excluded from participation in Congresses certain nations whose contributions to the progress of Science could not be overlooked. Two Congresses were held with these restrictions, one in Strasbourg in 1920, the other in Toronto, in 1924. But in the closing session of the latter, a motion by the representatives of the United States of America, seconded by the delegates from several other nations, including Italy, expressed the wish that the era of exclusions be ended.

The Council of the Union, shortly after, designated Bologna as the seat of the next Congress for the year 1928; a choice inspired by the renown of the old Italian town, the famous University of which soon will have existed for nine centuries.

It seemed that this choice could suggest an effective means to achieve what formed the wish of the majority of scholars: the return to an understanding in the field of our studies, which was no longer disturbed by painful memories and which delivered to science the recall of a state of mind that nothing more justified.

If the invitation to the Congress is made by the University of Bologna, and if the meeting takes place are the auspices of an Athenaeum which for centuries welcomed students from all over Europe; if this invitation is addressed to all those who cultivate the purest of all sciences; if the guests are not asked what nationality or school they belong to, but only if they value the progress of science and the benefits it brings, who can refuse membership, who will, for events that the current of history takes us further away from each day, perpetuating tangles where we only seek the consent of reason? This has been our thought, and that of the great majority of Italian scholars; in this sense, the invitation of the Rector of the University of Bologna was written, and the number and quality of the members that we are happy to see gathered in this historic room shows us that our way of acting, if it was able to derogate from some necessarily obsolete regulation, has obtained a consent which we can, not without pride, qualify as universal, and which some discordant voices, coming from the most opposing sides, only make more sensitive.

The exceptional man whom the fortune of Italy has brought up to direct its destinies has approved our line of conduct; the Congress had his support, like that of the representative of the government and the first magistrate of the city; thanks to this support, the Executive Commission of the Congress was able to fulfil its task, which was far from presenting itself as easy.

Royal Highness, Excellencies, Ladies and Gentlemen,

This Congress, to which His Majesty the King of Italy not only granted his high patronage, but where he wanted to be represented by an august prince of His House, His Excellency the Duke of Bergamo; whose Head of Government accepted the Honorary Presidency and to which he delegated His Excellency the Hon Belluzzo, Minister of Public Instruction, who is at the same time an eminent professor of one of the most important branches of applied mathematics; this Congress, we say, will begin its work. Works which are going to be considerable, because they include seventeen lectures, of a general character, delivered by scholars of high renown and relating to themes of greatest interest in various fields of pure and applied mathematics; more than 400 papers on the most varied subjects of Arithmetic, Analysis, Geometry, various branches of Mechanics and Engineering Science, Statistics, Actuarial Science and the Calculus of Probabilities, and Didactics; and while these papers will give rise to very interesting discussions, the historical section will highlight the contribution that Italy, and the Bolognese Athenaeum in particular, have made to science since the 15th century.

We dare to affirm that the memory of this meeting will be a milestone in the history of the development of scientific reports; we dare to believe that it opens a new series of Congresses, where the old misunderstandings will be forgotten, and where the scholars of all countries will periodically mark the progress obtained in this ideal domain which embraces the highest and most delicate associations of thought, and which maps out the technique for the directions to follow to contribute, by the most rational ways, to the well-being of humanity.
Then Professor Birkhoff, who speaks on behalf of the foreign members of Congress, gave, first in the English language then in a French translation, a speech that we report here in summary.

Speech by Professor G D Birkhoff:
Royal Highness, Gentlemen,

I address the greetings for your warm welcome from the mathematicians present here and representatives of forty nations from all parts of the world. We appreciate, more than we can express with words, the welcome reserved for us by the Italian government, the ancient and very famous city of Bologna and your University which is so justly celebrated.

It was quite natural for this International Congress to take place in Bologna and we feel honoured to be among you here, where our admiration for the splendid work of Italian mathematicians has drawn us back. The memory of the friendship and hospitality shown by you in the past in our regard is an incentive and a stimulus for us.

A memorable week of work is therefore heralded, and we extend our heartfelt thanks to you.
Lastly, His Excellency Belluzzo, Minister of Public Instruction, representative of the National Government, spoke.

Speech delivered by His Excellency G Belluzzo, Minister of Public Instruction:
Royal Highness, Gentlemen,

I am happy to bring to the illustrious mathematicians from all over the world gathered here at this Congress the greeting of the Fascist Government.

I think that it was not by chance that Bologna was chosen as the venue for this international conference: the mathematical traditions of the University of Bologna and the light of knowledge that it spread throughout the world are well worthy of welcoming such a select group of scientists.

The presence at this ceremony of a Prince of the glorious House around which all Italians gather with their hearts and minds increases its solemnity and importance.

Personally, I am honoured to represent here the Government which intends to give science the most dignified place, both as Minister of Public Instruction, and as an Engineer and Professor who has greatly benefited from mathematical science in studies, teaching and mechanical constructions.

For a long time I have been a warm advocate of the need for mathematical studies for engineers who want to do engineering science and I sometimes think, not without melancholy, of the huge sums that would have been spared in construction, which form the new wonders of the earth in different fields of technology, if the infinitesimal calculus had been more intimately known and severely used by the designers.

In fact, this technique is a highly effective weapon in mathematics, which allows it to solve the most complex and difficult problems and to always find solutions of maximum efficiency or minimum weight and dimensions, that is, of minimum expenditure.

There is no branch of technology, from mechanics to electrical engineering, from hydraulics to construction, which does not owe its progress to mathematical science: it is in fact mathematical studies that have made it possible to subject the large transmissions of the energy, to push the voltages of alternating electric currents, transmitted by being suspended or by wires imprisoned in cables, at values of thousands of volts; it is thanks to the results of mathematical studies on the critical speeds of the shafts, on the vibrations of the rotating masses that powerful machines can now be turned at the speed of thousands of revolutions per minute; it is because mathematics has made it possible to determine the laws of the disturbed motion of water in pressure pipes that water jumps of hundreds of meters can be used; it is because it taught us to calculate arches, vaults and beams that large structures of iron, of stone, of reinforced concrete were made economic and safe.

Mathematics is a formidable weapon that penetrates the areas in the shadow of knowledge and illuminates them; it is a powerful weapon that with physics and chemistry strives to demolish the wall behind which lies the mystery of creation, slowly and tenaciously conquering new positions, opening new vast horizons to experimental investigations.

Mathematics is the intimate collaborator of physics: where with its experience discovers new laws, highlights new phenomena, mathematics with its formulas consolidates the conquered position and extends it.

The great Maxwell has shown what mathematics can do in the mind of a genius, since he arrived with his formulas where physics only came later. It is mathematics that indicated to astronomers the existence and location of a new planet in the solar system. It is mathematics that has made it possible to calculate the infinitely large distances of planetary systems - those which are expressed in light-years - and the infinitely small distances of atomic systems which are expressed in fractions of a millionth of a millimetre. It is mathematics that has made it possible to calculate the mass of stars thousands of light-years away from the earth and that of the electrons that form an atomic system. It is mathematics that, once the hypothesis of the dependence of the mass of a body on its speed and the constancy of the speed of light has been established, has allowed us to create the theory of relativity and to arrive at conclusions that open to the astonished mind of men new unsuspected horizons.

On all the old and new roads, on which progress has travelled, mathematics has always been a powerful and effective engine.

James Watt invented the steam engine, but mathematics placed at the service of thermodynamics has perfected it, indicating the conditions necessary to increase its efficiency and therefore decrease fuel consumption. Telegraphic transmissions across the oceans were made possible by Lord Kelvin's mathematical studies.

The mission of mathematicians in the world is certainly among those most illuminated by ideals, since if the physicist or the chemist, discovering a new phenomenon or a new reaction, can sometimes turn the discovery to their own profit, while for the mathematician who manages to solve a complex problem that has occupied his mind for hours, days, weeks and sometimes years, there remains only the intimate satisfaction of the search made, of the solution found.

And it is certainly because mathematical discoveries only give intimate satisfaction and moral rewards that among mathematicians of the same nation and among those of different nations there exists a solidarity and a spirit of collaboration that is not always the rule in branches of applied science.

Of course, it can be painful to find that while the crowds lavish applause on millions of men who have the ability to exchange fisticuffs, and telephone and telegraphic lines and newspaper pages are placed at their disposal, men of science are ignored by the crowds and fleetingly remembered by the daily press when the cycle of their physical and intellectual life closes.

But it is good that it is so, because science is aristocracy, mathematical science is aristocracy among aristocracies, which must operate away from the crowds by descending silently and incognito to help improve or perfect their existence. Only history distinguishes those who with their minds have lived very close to the earth from men of science who have lived close to God in thought: it quickly forgets the former, to remember and glorify only the latter.

I do not know if a day will come when culture will be so widespread in the middle classes that they too can become aware of the efforts that have fatigued the intellect of mathematicians, of the nobility that has been the guide of these efforts, of the disinterest that it is the emblem, of the greatness which is its soul.

Wishing the next day in which the gratitude of the best will turn a mindful and grateful thought to those who were great mathematicians and will honour what they are, I allow myself as a technician and as a Minister, sure interpreter of the thought of all the technicians in the world, to make this act and to thank all of you for your contribution to progress in general and that of technology in particular.

With this thanks and with the most fervent wishes for the results of your works, in the name of Augustus His Majesty the King, I declare the International Congress of Mathematicians in Bologna open.
4.       Continuation of the Congress.

First plenary session.

At 15.00 on the same day, 3 September, the General Assembly of the Congress met in the main hall of the Chemical Institute to appoint the President and Vice-Presidents of the Congress.

The classroom was crowded by numerous members of the Congress. The members of the Executive Commission took their seats. The Secretary General, professor Ettore Bortolotti, asked the meeting to proceed with the appointment of the President of the Congress and the Vice-Presidents.

Professor Virgil Snyder, of Cornell University, asked to speak and proposed that Professor Salvatore Pincherle be appointed as President of the Congress. The proposal was approved by acclamation,

Professor Pincherle thanked the Assembly for the honour granted to him, and proposed that the Presidency, together with him, be constituted as follows:

Vice-presidents: Professor de la Vallée Poussin for Belgium - Professor J Hadamard for France - Professor D Hilbert for Germany - Professors W H Young and J C Fields for England and Dominions - Professor O Veblen for the United States of America - Professor H Fehr for Switzerland - Professor E Terradas for Spain and Latin America - Professor W Sierpinski for Poland - Professor H Bohr for Holland, Sweden, Norway, Denmark - Professor N Luzin for Russia and Ukraine - Professor S Kakeya for Japan.

General Secretary: Professor Ettore Bortolotti.

These proposals are approved unanimously by the Assembly.

The President proposed that telegrams of homage be sent to His Majesty the King, High Patron of the Congress and to His Excellency Benito Mussolini, Honorary President. The proposal is welcomed by the Assembly.

He also proposes that a greeting telegram be sent to Professor Émile Picard. The proposal is approved.

The President then gives the floor to Professor D Hilbert for the lecture: Probleme der Grundlegung der Mathematik.

The lectures of the professors follow: J Hadamard: Le développement et le rôle scientifique du Calcul fonctionnel. - U Puppini: Le bonifiche in Italia.

In closing the meeting, the Chairman proposed that Professor J Hadamard be the Chairman for the next session. The Assembly approves.

Tuesday 4 September.

At 9.00, plenary session.

Chaired by Professor J Hadamard. The Chairman announced that Professor Borel, who should have delivered the lecture with the title: Le Calcul des probabilités et les sciences exactes, is not present in Bologna and has sent the manuscript to be read in the Congress session. He therefore invites Professor E Cartan, who is present at the meeting, to read the announced lecture.

Professor E Cartan willingly accepts the assignment.

After this lecture, the President gives the floor to Professor O Veblen, who delivers the lecture: Differential Invariants and Geometry.

Then, the lecture of Professor G Castelnuovo: La Geometria algebrica e la Scuola italiana.

For the next plenary session, Professor de la Vallée Poussin is elected President.

At 16.00 the individual sections take place, the subdivision into Subsections, and the sessions in the Sections and Subsections begin.

Wednesday 5 September.

At 9.00, there was a plenary session chaired by Professor de la Vallée Poussin.

The Chairman introduced Professor W H Young, who delivers the lecture: The mathematical method and its limitations.

Following that, Professor V Volterra delivers the lecture: La teoria dei funzionali applicata ai fenomeni ereditari, and Professor H Weyl, the lecture: Darstellung kontinuirlichen Gruppen.

For the next plenary session, Professor W H Young is elected Chairman.

At 16.00 the Sessions in the individual sections begin, which continue until late in the evening.

At 21.30 the members of the Congress participate in the Reception offered in their honour by the Podestà of Bologna, Hon L Arpinati, in the municipal residence. The beautiful, large rooms of the ancient building of the Bolognese municipality are crowded with members of the Congress, their ladies, city authorities and representatives, who hold lively conversations until after 23.00.

The guests were served, with hospitable generosity, a sumptuous refreshment.

Thursday 6 September.

At 9.00, under the chairmanship of Professor W H Young, the plenary session takes place. The following lectures are delivered: V Kármán: Mathematische Probleme der modernen Aerodynamik - L Tonelli: Contributo italiano alla teoria delle funzioni di variabili reali. - L Amoroso: Le equazioni differenziali della dinamica economica..

After these lectures were given, Professor W Sierpinski is appointed as Chairman of the next plenary session.

At 16.00 sessions of the individual sections.

At 21.00, in the Teatro Comunale of Bologna, the orchestral concert of Italian music took place, conducted by Maestro Guarnieri, offered by the Organising Committee to the members of the Congress. The programme included selected musical pieces by famous composers of past and contemporary centuries, in order to highlight the characteristics of the various eras and highlight their most eminent qualities. The execution, in every aspect perfect, aroused in the numerous members of the audience frank enthusiasm and lively emotion.

Friday 7 September.

This day was dedicated to trips to Ravenna, Ferrara, Riva di Garda and Ponale, which took place according to the following programmes:

Programme of the trip to Ravenna.

7.00 - Departure from Bologna.

7.45 - Arrival in Imola - Caffè-latte

9.30 - Arrival in Ravenna.

9.45 - Reception in the Town Hall (Vermouth).

10.30 - Visit to the monuments.

12.15 - Departure from Piazza S Apollinare Nuovo to the Pineta di Classe.

13.15 - Lunch (offered by the Municipality of Ravenna in the Pineta).

15.30 - Departure for Porto Corsini.

17.30 - Departure from Porto Corsini for Ravenna.

18.00 - Departure from Ravenna to Bologna.

21.05 - Arrival in Bologna.

Programme of the trip to Ferrara.

7.00 - Departure from Bologna.

8.20 - Arrival in Pontelagoscuro - Visit to the works of the Boicelli Canal basin and the new aqueduct.

9.30 - Departure for Ferrara by car.

9.45 - Visit to the Duomo, the Cathedral, the Certosa, and the house of Ariosto.

11.30 - Reception offered by the Town Hall at the Palazzo dei Diamanti.

14.00 - Meeting in Piazza Castello.

14.15 - Visit to Ferrara University and the Library.

15.00 - Visit to Casa Romei, to Ludovico il Moro's Palace, to the Schifanoia Museum, etc.

17.30 - Departure from the station for Bologna.

18.20 - Arrival in Bologna.

Program of the trip to Riva del Garda and Ponale.

5.55 - Departure from Bologna.

8.30 - Verona - Caffè-latte.

10.00 Arrival in Rovereto.

10.15 - Departure from Rovereto to Riva with a special train of the Rovereto-Riva Consortium.

11.30 - Arrival in Riva.

11.45 - General explanation of the Ponale plant at the" Circolo Italia ".

12.15 - Lunch in Riva Hotels, offered by the Rovereto-Riva Consortium.

13.30 - Departure from Riva for Lake Ledro and the outlet works of the Ponale plant, with vehicles of the Rovereto-Riva Consortium.

16.00 - Visit to the Riva hydroelectric power station.

17.30 - Departure from Riva to Rovereto by special train.

19.30 - Departure from Rovereto.

23.25 - Arrival in Bologna.

The members of the Congress could participate, with a free choice, in one of these trips.

The Ravenna trip.

400 members of the Congress participated in the tour to Ravenna.

The pottery factories of Imola, on behalf of the Ordering Committee, had expressly manufactured the cup of caffè-latte, which was left, as a memento, to the members of the Congress.

In Ravenna, numerous road transport vehicles awaited the members of the Congress to take them to the town hall where the Podestà, Hon Calvetti, welcomed them on behalf of the city of Ravenna. Then His Excellency Hon Leicht, Undersecretary for Public Education and Professor at the University of Bologna, gave a speech offering the Congress a greeting on behalf of the Government. He said:
The members of the Congress, experts in the noblest of the sciences, will approach the city which contains many conspicuous memoirs, not only for lovers of art, but for profound thinkers. The image of Justinian, still alive in the mosaics of San Vitale, must have seemed throughout the Middle Ages almost a symbol of the present vitality of the empire, surviving the barbarian invasions. The divine poet thoughtfully paused before it, to whom the members of the Congress pay a noble homage. And it is right that the lovers of the most universal of sciences raise their thoughts to the one who was not only a great poet, but a great thinker and who was able to bring together, in a wonderful synthesis, all the knowledge of his time.

In front of the image of Justinian, before Dante's tomb, the members of the Congress will feel, better than anywhere, citizens of the ideal city that has neither towers nor doors, flooded by a sea of light and around which is a forest of virgin peaks and steep peaks: the harshly contrasted truth, the insidious truths of Dante's poem.
Professor Pincherle, President of the Congress, replied on behalf of the members of the Congress; he thanked the Podestà of Ravenna for the hospitable welcome, and, with a deep sense of emotion, the Head of Government, who wanted to be present at this event of the Congress, at the same time both happy and solemn, in the person, so dear to us our illustrious colleague.

Members of the Congress are offered an elegant toast of honour, then visits to the city's monuments begin. A long procession of road vehicles is therefore formed that lead the excursionists to the pine forest of San Vitale, where a sumptuous and lavish lunch offered by the Podestà of Ravenna have been set up on a table in a clearing among the ancient trees.

The treatment is perfect, the hospitality cordial and complete.

After the banquet, a series of trucks transported members of the Congress to Porto-Corsini, where a short stop allows them to see the sparkling and solemn Adriatic in the serene sunset.

The Ferrara trip.

Seventy-eight members of the Congress participated in the trip to Ferrara, accompanied by Professor U Puppini, professor of hydraulics and director of the engineering school of Bologna, who guided them in the visit to the important work in progress in the basin of the Boicelli canal, which will have to connect the Po with the Ferrara dock, and of the grandiose aqueduct intended to supply water to the whole municipality of Ferrara, deriving it from the Po, by means of a very modern filtration and purification system.

After the visit, the building company and the River Navigation Company offered lavish refreshments to the members of the Congress, who then, by means of numerous cars made available to them by the municipality of Ferrara, went to the city and were welcomed in the Castello Estense by the Vice-prefect and the President of the provincial deputation. They then visited the city monuments.

At 11.30, in the diamond palace, the Podestà of Ferrara, Mr Ravenna, addressed the members of the Congress welcoming them to Ferrara, recalling the glories of his university, which had the pride of hosting the immortal Copernicus.

Professor U Puppini replied on behalf of the members of the Congress.

The Podestà of Ferrara offered Congress participants a lavish refreshment.

The Riva trip.

Seventy-three members of the Congress participated in the trip to Riva di Garda and to the Ponale, which took place according to the established programme.

Arriving in Riva in the morning, they were received by De Francesco, Podestà of Rovereto, and Senator Conti, representative of the Autonomous Authority of Adige-Garda, and after a stop for breakfast, they visited the large Riva power station: then, travelling in cars, they headed towards lake Ledro, following the suggestive Ponale road that dominates lake Garda below them for a long stretch.

Once in Ledro, the group took a look at the project and the work done in order to exploit, by pouring the waters of Lake Ledro into Lake Garda, about 600 m. above it, and then visited the gallery under pressure and the two gate wells. After the visit and reluctantly saying goodbye to the small enchanting Lake, the members of the Congress returned to Riva, then to Rovereto, and in the evening they arrived in Bologna.

Saturday 8 September.

At 9.00, plenary session.

Chaired by Professor W Sierpinski, who immediately granted the floor to Professor M Fréchet, who delivers the lecture: L'analyse générale et les espaces abstraits. Then the following lectures take place: R Marcolongo, Leonardo da Vinci nella storia della matematica e della meccanica. - N Luzin, Sur les voies de la théorie des ensembles.

After the morning of the Congress, the members of the Congress met at the Littoriale, where the 1100-seat lunch offered by the Organising Committee took place.

At the table of honour, in addition to the President of the Congress Professor Pincherle and the President of the Committee Professor G Albini, Rector of the University, sat the Prefect of the province His Excellency G Guadagnini, the Podestà of Bologna, Arpinati, the civil and military authorities of the city and province and delegates of Foreign Countries to the Congress.

With champagne Professor S Pincherle pronounced high words of thanks to the representatives of the Government and the Municipality, and expressed the satisfaction of the members of the Congress for all the excellent success of the Congress.

The lunch took place between the maximum liveliness, and, despite the exceptional number of those present, with perfect order.

At 16.00, last session in the individual sections and subsections took place.

5.       Closing Session of the Congress.

At 22.00 the members of the Congress were welcomed in the splendid halls of the government building by His Excellency the Prefect G Guadagnini, who offered them a solemn reception marked by cordial elegance.

The vast halls appeared crowded; and animated conversations were held between the major personalities of politics and science with foreign professors and the ladies who came in their company to visit Italy. At 23.00 the buffet room was opened; the reception lasted until late, and left the best impression in the minds of the members of the Congress, who wanted to express to His Excellency the Prefect their deep gratitude for the welcome received, for the facilities that the political and city authorities have granted them and for the warm, spontaneous hospitality offered to them by citizens, such as to leave them dear and grateful memories of their stay in Bologna.

Sunday 9 September.

The plenary session arranged for 9.00 could not take place due to the absence of Professor F Enriques, who should have delivered the lecture: Continuità e discontinuità nella Geometria algebrica.

Meeting of the International Mathematical Union.

The Assembly of the International Mathematical Union was held in one of the classrooms of the Mathematical Institute on 9 September at 10.00.

The following countries belonging to the Union were represented: Belgium, Canada, Czechoslovakia, Denmark, France, Japan, Great Britain, Italy, Holland, Poland, United States of America, Sweden, Switzerland (13 countries out of 19).

The following members of the presidency were present: de la Vallée Poussin and Fields, Presidents of honour; Pincherle, President-in-Office; Young and Fehr, Vice-Presidents. Chaired by Professor S Pincherle, President with Professor H Fehr acting as Secretary.

In opening of the meeting, the President warned that the meeting could only have an unofficial character, since the Secretary General, due to a matter of principle, did not consider it appropriate to convene the assembly of delegates. He presented the minutes of the meeting at Toronto.

He then explained the difficulties that arose in the organisation of the Congress, to which the necessity was imposed that mathematicians from all countries be invited, without exceptions whatsoever.

To overcome those difficulties, the Congress was placed under the auspices of the University of Bologna, and the invitations were made by the magnificent Rector of that University, a scientific authority universally recognised and free from any political significance. In this way the Congress was able to resume the truly international character that all the Congresses of the series inaugurated in Zurich in 1897 had had, and continued until the beginning of the great war.

The assembly approved, by unanimous votes and with acclamation, the following agenda, which established the line of conduct taken by the President in the preparation of the Congress:

The members of the International Mathematical Union are very grateful to Professor S Pincherle for what he has done for the success of the Bologna Congress, and they approve entirely. To study the current situation, they refer to the Presidency of the International Mathematical Union.

Professor Pincherle thanked the meeting for this vote of confidence, believed however that he cannot keep the office of President of the International Mathematical Union, and asks the Presidency to take note of his resignation, which was absolutely irrevocable. After that, he left the President's seat to the Honorary President, Mr de la Vallée Poussin.

The second part of the session was devoted to an exchange of views on the choice of the venue for the next International Congress of Mathematicians, as a preparation for the session of the general assembly of the members of the Congress to be held in Florence on day ten.

Honouring dal Ferro and Cavalieri.

At 16.00 the members of the Congress saw the unveiling of a plaque placed on the paternal house of Scipione dal Ferro and of another plaque placed on the facade of the church of Mascarella where Bonaventura Cavalieri was prior.

Professor Bortolotti briefly recalled the periods of Bolognese mathematical history in which those two mathematicians flourished, and explained what meaning they have in the development of algebraic theories and in the foretaste of infinitesimal analysis.

Monday 10 September.

At 6.25 those Congressmen who had not already left for Florence took their seats in a special train that took them to that city where they arrived at 9.20.

At 11.00 they all gathered in Palazzo Vecchio, and in the hall of the Cinquecento, with the most solemn appearance, the Assembly of the members of the Congress took place.

The session began with fervent words of greeting from the Podestà of Florence, Mr Garbasso, who made the following speech:
Excellencies, Ladies and Gentlemen,

I am happy and proud to extend to the members of the International Congress of Mathematicians the greeting and homage of the Municipality and people of Florence.

And I am happy to be able to express to the Executive Commission and its illustrious President in particular, Professor Salvatore Pincherle, our deep gratitude. It is a great honour for us that the Congress closes its noble effort by meeting for the last time in this old house of the Florentine people. We acknowledge, however, if you allow me, without false modesty, that nowhere could perhaps have been held more worthily than here, the lecture by Professor Birkhoff: Quelques éléments mathématiques de l'art.

You know, ladies and gentlemen, that if Florence and Tuscany were, since the Middle Ages, the cradle of the arts, they have also made a contribution of the highest order to the renaissance of mathematics.

I will have the pleasure, soon, to show you the scrolls of our scientific nobility.

His Excellency the Minister of Education wanted to authorise the meritorious directors of the National Library and the Laurentian Library, Professors Bruschi and Rostagno, to exhibit here the documents proving our birthright.

It was a Tuscan, Leonardo Fibonacci of Pisa, who at the beginning of the thirteenth century revealed the algebra of Indians and Arabs to Christian Europe. His "Liber abbaci" was composed in 1202.

It was a Tuscan, Raffaello Canacci, of Florence, who, in the fourteenth century, published the first treatise on algebra that was written in a modern language.

Leonardo da Vinci was a Tuscan who at the end of the fifteenth century devised the new mechanics.

It was a Tuscan, Galileo Galilei, who founded modern astronomy and rational mechanics at the end of the sixteenth century and the beginning of the seventeenth.

Galileo Galilei was also the first who, after Archimedes, did what we now call an integration. Three disciples of Galileo: Bonaventura Cavalieri, Evangelista Torricelli and Vincenzo Viviani were among the first to found infinitesimal calculus. If the discovery of the general methods made us forget its precursors, the results of Cavalieri, Torricelli and Viviani are mentioned, however, on almost every page, in Isaac Newton's Nova methodus fluxionum, and this is enough for the glory of our Italians.

When we quickly summed up the contribution that Florence made to the development of the most perfect of sciences, the influence that mathematics exercised in the formation of the elegant, sober and limpid genius of the Tuscan people was not yet highlighted.

Our libraries conserve tens of arithmetic, algebra and geometry treatises, composed in the 14th and 15th centuries for those merchants who were then the bankers of the popes, emperors, kings of France, and king of England.

And our libraries retain numerous essays on mathematics and mechanics, due to artists called Lorenzo Ghiberti, Leone Battista Alberti, Antonio Filarete, Francesco di Giorgio Martini and Leonardo da Vinci. In Florence, as in ancient Greece, mathematics was therefore a very important element of civilisation.

But in Florence, as in ancient Greece, the greatest flowering of the arts preceded the maximum flowering of the sciences.

And if in the gardens of Academy the wise men fixed the immortal truths in the shadow of the immortal statues, it seems it was necessary that Sandro Botticelli rediscovered perfect grace and Raffaello Sanzio, here in Florence, perfect beauty, because Galileo Galilei could finally reveal the thought of Plato's geometer god.

But I realise, Ladies and Gentlemen, that I am giving you a speech on the artistic elements of the sciences, a speech that is not included in the programme compiled by my illustrious colleague, Professor Pincherle.

I apologise for having diverted you and please allow me only to express the hope that you will be able to bring back from your visit to Florence the memory of a city that was not only, as the guides say, "une ville d'art célèbre", but indeed, and at all times, one of the bright lights of universal civilisation."
After this Professor Birkhoff, of Harvard University, gave his lecture on the theme: Mathematical elements in art.

The distinguished American scientist demonstrated with original concepts how mathematical proportion is fundamental in art, which is then a continuous search for geometric symmetry.
To truly understand a work of art - he says - one must first grasp its object, making an effort proportional to its complexity to have an explicit perception of the more or less hidden symmetry and harmony.
Based on this principle, Professor Birkhoff raised the question of knowing to what extent the quantity of order relations that exist between the elements of an artistic object has a relationship with the complexity of the object itself.

A problem that he subtly solved by reaffirming the importance of the geometric factor in the understanding of an artistic work, since the aesthetic expression varies with varying dimensions.

After the conference, Professor Pincherle, President of the Congress, thanked the Podestà on behalf of the members of the Congress, then reported on the preliminary decision made for the designation of the location of the future Congress, and, as a result of that decision, put to the vote the proposal that the next international Congress should take place in Switzerland in 1932.

After a cordial agreement by the Swiss Delegate Professor Fueter, the proposal was approved by unanimous acclamation.

Professor Pincherle noted with great satisfaction this manifestation of the General Assembly of the members of the Congress, and ended by hoping that the new bonds of camaraderie will be tightened among scientists, for an ever-growing development of science and civilisation.

After the session was closed, the members of the Congress went to visit the very interesting exhibition of rare mathematical science books, displayed in the Palazzo Vecchio by the director of the National Library Mr Bruschi, and from that of the Laurentian Library Mr Rostagno.