# 1900 ICM - Paris

## 1900 International Congress of Mathematicians - Paris, France

The International Congress of Mathematicians was held in Paris, France from 6 August to 12 August 1900. The Congress was attended by 250 full members. We give below a version of:
Before presenting the material, we give a short Preface.

**Preface by EFR and JJOC.**

In many ways it was David Hilbert's lecture on the future problems of mathematics which became the most famous lecture delivered at any International Congress of Mathematicians. It is sometimes described as a plenary lecture and sometime it is not. We explain why there is a confusion. Hilbert was invited to give a plenary lecture at the 1900 Congress. He could not decide on a topic and discussed what he should talk on with Hermann Minkowski and Adolf Hurwitz. Eventually he decided to discuss outstanding mathematical problems but took a long time to write his lecture. The programme for the 1900 Congress was published before he had things ready and so his talk was not advertised as a plenary lecture. It was delivered at the Congress in a joint session of the Teaching Section and of the History Section. By the time the Proceedings of the conference appeared the organisers decided that, because of its importance, they would put Hilbert's paper in the section with the plenary lectures. Another point worth making is that, because of lack of time, Hilbert only spoke about 10 of his 23 problems in the lecture although all 23 appear in the published paper. Most of this Congress was in French, but there was a discussion about the problems of mathematics being published in such a diverse number of languages.

**1. Summary report of the Congress.**

The second International Congress of Mathematicians was held in Paris from 6 to 12 August 1900; we know that the Zurich Congress had entrusted the organisation to the French Mathematical Society.

The Universal Exhibition, which took place in Paris, was so attractive in itself that it would have been difficult to organise successfully for members of Congress special excursions, as had been done in Zurich. The Organising Committee thought it preferable to leave the delegates complete freedom, and had to confine itself to a few meetings, apart from the sessions proper.

The first of these meetings took place at the Café Voltaire, on Sunday 5 August, at 20:30 in the evening; before the official opening of the Congress, it allowed many of its members to make contact, to link up or to renew acquaintances.

The delegates gathered on Tuesday, 7 August for a lunch, served at the École Normale Supérieure whose Director, M Perrot, had been so kind as to provide us with the Salle des Actes; Mrs Jules Tannery has graciously agreed to do the honours. MM Perrot, Moritz Cantor and Darboux took the floor successively in toasts which were greatly applauded.

The delegates were received on Friday 10 August by the President of the Republic, and on Saturday 11 August, at the same time as their colleagues at the Congress of Physics, by Prince Roland Bonaparte, whose generous dedication we know to scientific enterprises.

Finally, after the close of the Congress, its members gathered in large numbers on Sunday 12 August, at noon, in a farewell banquet, which many ladies and girls were kind enough to honour and charm with their presence. The President of the Congress unfortunately found himself too tired to take part in it. Many toasts were brought, first by M Darboux, who excused M Poincaré, then by M Geiser, on behalf of foreign delegates, by M J Tannery who proposed the health of those absent, and by MM Stephanos and Vassilief; M Darboux, finally, spoke again in his own name, and improvised a charming speech, in which he highlighted how much it is necessary for mathematicians to feel close, since there are no scholars whose work is more disinterested, nor less likely to conquer the masses of laymen.

Some members of congress met again in the evening at the Opera, at a gala evening for which the Minister of Public Education and Fine Arts had kindly reserved a few places for our Congress.

The reports of the general meetings and those of the sections are given below; with the exception of the opening session held at the Palais des Congrès, located within the Exhibition grounds, they took place at the Sorbonne, which the Rector of the Academy of Paris had graciously put at our disposal the amphitheatres Cauchy, Le Verrier and Chasles, of the Faculty of Sciences, as well as the Richelieu amphitheatre, for the general closing session.

**2. Opening Session.**

It was at the Palais des Congrès, built on the grounds of the Universal Exhibition, that the opening session was held on Monday 6 August, from nine in the morning to half past eleven. In the audience, many ladies and young girls with clear faces.

M Jules Tannery, on behalf of the Minister of Public Education and Fine Arts, of which he is the official representative, declared the Congress open and welcomed foreign members. He proposed to entrust the presidency of the Congress to M Henri Poincaré, who was appointed by acclamation.

M Poincaré thanked the assembly and proposed to appoint honorary president M Hermite, who was prevented from coming in person to attend the work of the Congress because his advanced age, but who, nevertheless, explained M Poincaré, "is with us in heart". This proposal was greeted with prolonged applause.

Continuing his brief speech, the President recalled that, among the resolutions to be taken by the Congress, was one which will fix the date and place of the next Congress, as well as the bodies or associations responsible for preparing and organising it. He mentioned that, during the first Congress, many mathematicians had expressed the wish that the third Congress be held in Baden-Baden.

Finally, on the proposal of the President, the following were appointed by acclamation:

Vice-Presidents: MM Czuber, Gordan, Greenhill, Lindelöf, Lindemann, Mittag-Leffler, Eliakim Moore, Tikhomandritzky, Volterra, Zeuthen.

Secretary General: M Duporcq.

Secretaries: MM Bendixson, Capelli, Minkowski, Ptaszycki, Whitehead.

Chairmen of the six sections: MM Hilbert, Painlevé, Darboux, Larmor, Prince Roland Bonaparte, Moritz Cantor.

Section secretaries: MM Cartan, Hadamard, Niewenglowski, Levi-Civita, d'Ocagne, Laisant.

In accordance with the programme, the President gave the floor to M Moritz Cantor, who read in French his lecture

*On the Historiography of Mathematics*; then to M Vito Volterra, who read, also in French, his lecture to conference entitled:

*Betti, Brioscia, Casorati*.

M Rados then spoke a few words on behalf of the Minister of Public Education of Hungary, of which he is the official delegate. Finally, the Secretary General provided some oral indications to the delegates.

The meeting rose at 11:30.

**3. Timetable for the Congress.**

*Sunday 5 August*.

20.30 Preparatory meeting at the Voltaire café.

*Monday 6 August.*

9.00 General opening session at the Palais des Congrès.

*Tuesday 7 August.*

9.00 Sessions of sections I and II.

14.00 Sessions of sections III and IV.

16.00 Lunch at the École Normale Supérieure.

*Wednesday 8 August.*

9.00 Sessions of sections V and VI.

14.00 Sessions of sections V and VI.

*Thursday 9 August.*

9.00 Sessions of sections I and II.

14.00 Sessions of sections III and IV.

*Friday 10 August.*

9.00 Sessions of sections V and VI.

16.00 Reception at the Elysée.

*Saturday 11 August.*

9.00 Closing session.

21.30 Reception at the home of Prince Roland Bonaparte.

*Sunday 12 August.*

11.30 Banquet, hall of the Athénée Saint-Germain.

20.30 Gala evening at the Opera.

**4. Sessions of the Sections.**

**Section I. - Arithmetic and Algebra.**

*Tuesday 7 August.*

Chairman M Hilbert. The session is opened at 9.00 at the Cauchy Amphitheatre, of the Faculty of Sciences, chaired by M Hilbert. Mr Cartan performed the duties of Secretary. M Hilbert opened the session with a short speech in which he underlined the importance of Arithmetic and Algebra, which constitute grammar of Mathematics.

Communications:

- L Autonne, Sur les groupes d'ordre fini contenus dans le groupe linéaire quaternaire.

M Hilbert made a few comments on this subject.

- H Hancock, Remarks on Kronecker's modular systems.

M Hilbert made a few remarks on this subject.

- C Stephanos, Sur la séparation des racines des équations algébriques.

This Communication was followed by some observations by the President and an exchange of ideas between M Stephanos and M André, especially on certain theorems that M Stephanos thought new and that M André had already announced.

- H von Koch, Sur la distribution des nombres premiers.

M Hilbert presented some personal ideas on this subject and highlighted the importance of the results obtained by M von Koch. The meeting was adjourned around 11.00.

*Thursday 9 August .*

Chairman M Hilbert.

The session is opened in 9.00 at the Cauchy Amphitheatre.

Communications:

- R Perrin, Sur les propriétés d'un certain covariant de la forme binaire du cinquième ordre et leur application à la résolution de l'équation.

At the request of M Gordan, the author repeated a few points from his communication.

- L E Dickson, The known systems of simple groups and their inter-isomorphims.

Since MM Rados, Artemas Martin and Padoa, whose communications were on the agenda, were absent, the meeting was adjourned at 10.30.

*Friday 10 August*.

The following Communication from M Padoa, relating to the first Section, could have been made at a meeting of Sections V and VI:

- A Padoa, Un nouveau système irréductible de postulats pour l'Algèbre.

**Section II - Analysis.**

*Tuesday 7 August.*

Chairman M Painlevé.

The session is opened at 9.00 in the Le Verrier Amphitheatre. M Hadamard acted as Secretary.

Communications:

- M A Tikhomandritzky, Sur l'évanouissement des fonctions thêta de plusieurs variables.

- G Mittag-Leffler, Sur une extension de la série de Taylor.

MM Borel, Hadamard and Painlevé present some remarks on the previous communication.

*Thursday 9 August.*

Chairman M Painlevé.

- I Bendixson, Sur les courbes définies par les équations différentielles.

Father Issaly took advantage of the opportunity offered by M Bendixson's communication to talk about pseudo-surfaces. M Hadamard made an observation on the indication provided by the result obtained by M Bendixson, for the case where the right-hand side of the differential equation is of the third degree. He believed that in the absence of a rigorous demonstration this result only allows to a certain extent to form an opinion on what happens when the degree of this right-hand side is arbitrary. The questions relating to differential equations seem to be, in fact, those in which we can least trust these kinds of inductions. Thus, for geodesics of surfaces with opposite curvatures, as long as the connection order is equal to 1 or 2, the conclusions are very simple and do not differ from those given in the discussion relating to hyperboloid of one sheet. On the contrary, as soon as the connection order is at least equal to 3, the results change entirely in nature and become relatively complicated.

- E Jahnke, Nouveaux systèmes orthogonaux pour les dérivées des fonctions thêta de deux arguments.

- J Drach, Sur l'intégration des équations aux dérivées partielles du second ordre.

- H Padé, Aperçu sur les développements récents de la théorie des fractions continues.

- E Cartan, Sur l'intégration des systèmes d'équations aux différentielles totales.

**Section III. - Geometry.**

*Tuesday 7 August.*

Chairman M G Darboux.

The session is opened at 14.00 at the Cauchy Amphitheatre. M Niewenglowski acted as Secretary.

Communications :

- E-O Lovett, Sur les transformations de contact entre les lignes droites et les sphères.

- C Stephanos, Belations entre la Géométrie projective et la Mécanique.

M Massau recalled a Memoir which he published a few years ago on similar questions.

- A Macfarlane, Application of space analysis to curvilinear coordinates.

The session is adjourned at 16.00.

*Thursday 9 August.*

Chairman M G Darboux.

- F Amodeo, Coup d'oeil sur les courbes algébriques au point de vue de la gonalité.

- I Stringham, Orthogonal transformations in elliptic, or in hyperbolic space.

- V Jamet, Sur le théorème de M Salmon concernant les cubiques planes.

- A Padoa, Un nouveau système de définitions pour la Géométrie euclidienne.

M Vassilief presented some comments on this communication.

- Abbé Issaly, Sur les pseudo-surfaces en général et sur un exemple particulier de pseudo-surfaces minima.

**Section IV. - Mechanics.**

*Tuesday 7 August.*

Chairman M E Czuber.

The session is opened at 14.00 at the Le Verrier Amphitheatre. M Levi-Civita acted as Secretary.

Communications:

- I Fredholm, L'inversion des intégrales définies et son application aux problèmes de la Physique mathématique.

*Thursday 9 August.*

Chairman M Larmor.

- J Boccardi, Sur le calcul des perturbations spéciales des petites planètes.

- G Sousloff, Sur le mouvement d'un corps solide autour d'un point fixe.

Since MM Zenger and Somigliana, whose communications were on the agenda, were absent, the Chairman declared the work of the Section finished.

*Friday 10 August.*

MM Hadamard and Volterra nevertheless made, on Friday morning, at the sessions of Sections V and VI, the following Communications to be attached to the Mechanics Section:

- J Hadamard, Sur les équations aux dérivées partielles à caractéristiques réelles.

- V Volterra, Sur les équations aux dérivées partielles.

**Sections V And VI. - Bibliography And History. - Teaching And Methods.**

*Wednesday 8 August.*

Chairman M M Cantor.

First session.

Due to the absence of the Chairman of the Fifth Section, Prince Roland Bonaparte, Sections V and VI meet under the chairmanship of M Cantor, President of the Sixth Section. MM d'Ocagne and Laisant served as Secretaries.

The session was opened at 9.00 am.

Communications:

- D Hilbert, Sur les problèmes futurs des Mathématiques.

M Peano said that M Padoa's subsequent communication would address M Hilbert's problem # 2. M Mehmke recalls that he proposed certain monographic representations in space from which could result a solution of the general equation of the seventh degree.

- R Fujisawa, Note on the Mathematics of the old Japanese school.

M A Vassilief asked if we could not find traces of Greek influence, through the Greco-Bactrian kingdom, on the first Japanese geometers.

- L Leau, Proposition d'un voeu pour l'adoption d'une langue scientifique universelle.

M Leau wished to propose to Congress the following:

(a) It is necessary to adopt a universal scientific and commercial language.

(b) The official Academies are respectfully invited to agree on the realisation of this project.

Mr Leau also moved the following resolution:

The Congress decides to appoint five Members to the Delegation which is formed:

(a) To ask the official Academies to be good enough to adopt a universal auxiliary language;

(b) To make this choice itself or by a Committee created by it, in case the Academies refuse.

The President, because of the importance of these decisions, proposed to postpone the discussion to the Friday meeting, in order to give the delegates time to exchange their views.

Before separating, we decide to continue the work of the Section at 14.00. The meeting closed at 11.15.

Second session.

The session is opened at 14.00.

Communications:

- M d'Ocagne, Sur les divers modes d'application de la méthode graphique à l'art du calcul.

- Z de Galdeano, Note sur la critique mathématique.

M Vassilief made a remark on this subject.

- A Capelli, Sur les opérations fondamentales de l'Arithmétique.

A discussion began on the question between MM Capelli and Padoa.

- M Laisant placed on the desk a few copies of an article from the Bibliotheca mathematica on the state of progress of the work of the Bibliographic Directory of Mathematical Sciences.

- M Laisant presented the first booklet of a French-German and German-French mathematical vocabulary, by M Félix Müller.

The meeting closed at 16.00.

Friday 10 August.

Chairman M C F Geiser.

The session is opened at 9.00.

It begins with the communication from M Padoa, attached to the first Section, and those from MM Hadamard and Volterra, which relate to the fourth.

It is continued by the following communications:

- M Laisant read a Note from M Ch Méray, Professor at the University of Dijon, Sur la langue internationale auxiliaire de M le Dr Zamenhof, connue sous le nom de "Esperanto".

- The meeting continued with the discussion relating to M Leau's proposals, presented in the first session of the Section and relating to the adoption of a universal language, a discussion which, as we have seen previously, had been deferred to the current meeting. A discussion took place between MM Couturat, Laisant, Leau, Schröder and Vassilief.

M Vassilief believed that the movement which aims to introduce a universal language is very pleasing, and has great reasons for being so, owing to the rapidly growing variety of languages which are now used by scholars for the publication of their works. While at the beginning of the 19th century it was enough for a scientist to know Latin, French and English (Gauss wrote in Latin), the number of languages used in scientific literature increased as a result of national movements and continues to grow. For example, in recent years, interesting Mathematical Memories have been published in the Serbian, Croatian and Ukrainian languages. The great passion with which the Japanese pursue scientific studies will bring us face to face with Japanese literature if Japanese scholars no longer want to write in other languages in the great interests of science as they have done so far. Ten, twenty or thirty scientific languages will be a great danger for science; nevertheless, M Vassilief thinks that it is not an artificial language, such as the Esperanto language, or a dead language like Latin, which can replace living languages, which are due to the work of several generations of thinkers and writers. It is therefore, according to him, in another way that we must seek the solution of the problem posed by M Leau's proposals. It is necessary to express the desire that publications made in little known languages be followed by abstracts written in a language better known in the scientific world, as do now the Academies of Copenhagen, Krakow, and the Mineralogical Society of St Petersburg; the complete works of first-rate scholars must be published in two languages, as did the Academy of St Petersburg for the Works of Chebyshev, that the Academies favour works such as the French-German Mathematical Dictionary, by Félix Müller, or the publication of the Mathematical Encyclopaedia of Meyer and Burkhardt in French. Consequently, M Vassilief proposed to express the following wish, more general than that of M Leau, and which can have more practical consequences:*that the Academies and Learned Societies of all the countries study suitable means for remedying the evils which come of the growing variety of languages used in scientific literature*.

- M R Guimaraes placed on the desk some copies of a memoir
*On Mathematics in Portugal in the 19th century*, an historical and bibliographic memoir that M Guimaraes had been commissioned to write for the Universal Exhibition.

In this monograph, he makes a rapid examination of the works of his compatriots, published during the nineteenth century, following in this study the classification, the notation and the abbreviations adopted by the Commission of the Bibliographic Directory of Mathematical Sciences. There are reports of 769 works, of which 226 belong to Analysis, 192 to Geometry and 351 to Applied Mathematics.

- E Maillet, Sur l'utilité de la publication de certains renseignements bibliographiques en Mathématiques.

MM Dickstein and Laisant make some remarks on this subject.

- M Vassilief informed the Section that Professor D M Sintsov published in the Bulletin of the Physico-Mathematical Society of Kasan (Vol. VII, VIII, IX), under the title: Bibliographica mathematica rossica, the detailed and systematic list of works and mathematical articles published in Russia in the years 1896, 1897 and 1898.

**5. Closing Session.**

The closing session opened on Saturday 11 August at 9.00, at the Richelieu amphitheatre, under the chairmanship of M Poincaré.

It is first of all a question of agreeing on the preparation and the date of the next Congress. We decide that it will take place in 1904 and we charge the German Mathematical Society to organise it and to choose itself the city where it will be held.

On behalf of several Members of the Congress, M Laisant proposed to send the following telegram to M Hermite, on vacation in Saint-Jeande-Luz:

The International Congress of Mathematicians sends the expression of its admiration and its respectful sympathy to the illustrious mathematician who honours his country and the whole scientific world by his talent as well as by his character. It is the unanimous wish of mathematicians of all nations to send M Hermite the most sincere wishes of happiness and health.M Laisant's proposal was accepted by acclamation.

Note: On receipt of this dispatch, M Hermite replied with the following telegram, addressed to M Poincaré, but which unfortunately arrived too late to be able to make it known to the Members of Congress:

Please be the interpreter of my feelings of gratitude to the Members of Congress and express to them how deeply I am touched by the expression of their sympathy; it reaches me at the end of my career, it is the highest and the best reward, it fills me with joy and honour by attaching the bonds of affection to those of Science; I answer with all my heart by thanking the friends I owe to my study, by sending them my wishes for the success of the Congress, so that it will support their work and contribute to the magnificent future of Analysis in new ways that it opened.The members of Congress then voted to thank M Gréard, rector of the University of Paris, for the hospitality he kindly extended to them at the Sorbonne.

Charles Hermite

The session ended with the lecture of M Henri Poincaré,

*On the role of intuition and logic in Mathematics*, and that of M G Mittag-Leffler, entitled:

*A page from the life of Weierstrass*.

At 11:30 the President declared the work of the Congress closed.