## 1893 International Mathematical Congress - Chicago

The World's Columbian Exposition was held in Chicago in 1893. The University of Chicago had been founded in 1892 and the new Faculty of Mathematics organised an International Mathematical Congress, as part of the Exposition, to be held from 21 August to 26 August 1893. We present below:

- The Title page of the Proceedings.

- The Preface to the Proceedings.

- Preliminary Arrangements.

- Formal Opening.

- 'The Present State of Mathematics', by Felix Klein.

- Organisation and Programme.

- The final session.

**Preface by EFR and JJOC.**

Felix Klein gave the opening address at the International Mathematical Congress held in conjunction with the World's Columbian Exposition in Chicago in 1893. The address, 'The Present State of Mathematics', contained a "manifesto" for future international cooperation by mathematicians. Klein, like Georg Cantor whose efforts had led to the founding of the German Mathematical Society, both strongly believed in international collaboration in mathematics. Their aims were the same but both were motivated by different personal reasons. Cantor felt his close colleagues were making unfair criticism of his work so he wanted a broader platform to promote his ideas. Klein had strong ideas about teaching mathematics and mathematical research and was a great organiser who wanted to see his ideas placed on a broader stage. This Congress is an important step in initiating the series of International Congresses of Mathematicians. If you look at the papers published in the Proceedings of this 1893 Congress, it appears to have been attended by many leading European mathematicians. This, however, was not the case for only 4 of the 45 participants were not from the United States. Klein brought papers from German, French, Austrian, Italian and Swiss mathematicians for publication in the Proceeding.

**1. Title page of the Proceedings of the 1893 Congress.**

Papers Published by the American Mathematical Society

MATHEMATICAL PAPERS READ AT THE INTERNATIONAL MATHEMATICAL CONGRESS HELD IN CONNECTION WITH THE WORLD'S COLUMBIAN EXPOSITION CHICAGO 1893.

Edited by the committee of the congress

E Hastings Moore

Oskar Bolza

Heinrich Maschke

Henry S White

New York, Macmillan And Co.

for the

American Mathematical Society 1896.

**2. Preface to the Proceedings of the 1893 Congress.**

The Mathematical Congress of the World's Columbian Exposition - of whose proceedings a brief report follows this preface - entrusted the publication of the papers presented to the Congress to the Chicago committee editing this volume. Neither the management of the Exposition nor the government of the United States had made provision for the publishing of the proceedings of any of the Chicago Congresses. No publisher was found willing to issue the papers at his own risk.

At last a guaranty fund of one thousand dollars in all was subscribed, six hundred dollars by the American Mathematical Society, and four hundred dollars by members of that Society and other mathematicians. On the basis of this guaranty fund the publication of the volume of papers was made possible, the American Mathematical Society assuming the financial, and the Chicago committee the editorial responsibility.

The Editors take this opportunity to express their grateful appreciation of the generosity of the subscribers to the guaranty fund, and of the interest in the undertaking shown by the officers of the American Mathematical Society. They desire also to thank Messrs Macmillan and Co. for the satisfactory dress in which the papers appear.

The Editors

Chicago, December 1895.

**3. Preliminary Arrangements for the Congress.**

In the schedule put forth by the World's Congress Auxiliary of the World's Columbian Exposition of 1893, the week beginning on the twenty-first day of August was designated for Congresses on Science and Philosophy. Early in 1893 the local committee for the Department of Mathematics and Astronomy had sent invitations to a large number of eminent specialists in those sciences in American and European countries. In response to these invitations, many contributions were received by the local committee before the opening of the Congress. The government of one country, Germany, had delegated an Imperial Commissioner to attend the Congress in person, Professor Felix Klein of Göttingen, who brought nearly all the mathematical papers contributed by his countrymen, and cooperated effectively with the local committee in the preliminary arrangements.

**4. Formal Opening of the Congress.**

The general session of all congresses in the Department of Science and Philosophy, convened in the Memorial Art Palace, Hall of Columbus, at 10.30 a.m. of Monday, 21 August 1893. After an address of welcome by Mr Charles C Bonney, President of the World's Congress Auxiliary, responses were made by foreign delegates. The assembly then dispersed, to meet immediately in the smaller rooms set apart for the several divisions.

**4.1. Mathematics and Astronomy.**

The divisions for Mathematics and Astronomy convened in Room 24 at 12 noon, under the chairmanship of Professor G W Hough of Northwestern University. After the introductory address of the chairman, Professor Klein addressed the division upon "The Present State of Mathematics."

**5. The Present State of Mathematics, by Felix Klein.**

The German Government has commissioned me to communicate to this Congress the assurances of its good will, and to participate in your transactions. In this official capacity, allow me to repeat here the invitation given already in the general session, to visit at some convenient time the German University exhibit in the Liberal Arts Building.

I have also the honour to lay before you a considerable number of mathematical papers, which give collectively a fairly complete account of contemporaneous mathematical activity in Germany. Reserving for the mathematical section a detailed summary of these papers, I mention here only certain points of more general interest.

When we contemplate the development of mathematics in this nineteenth century, we find something similar to what has taken place in other sciences. The famous investigators of the preceding period, Lagrange, Laplace, Gauss, were each great enough to embrace all branches of mathematics and its applications. In particular, astronomy and mathematics were in their time regarded as inseparable.

With the succeeding generation, however, the tendency to specialisation manifests itself. Not unworthy are the names of its early representatives: Abel, Jacobi, Galois and the great geometers from Poncelet on, and not inconsiderable are their individual achievements. But the developing science departs at the same time more and more from its original scope and purpose and threatens to sacrifice its earlier unity and to split into diverse branches. In the same proportion the attention bestowed upon it by the general scientific public diminishes. It became almost the custom to regard modern mathematical speculation as something having no general interest or importance, and the proposal has often been made that, at least for purpose of instruction, all results be formulated from the same standpoints as in the earlier period. Such conditions were unquestionably to be regretted.

This is a picture of the past. I wish on the present occasion to state and to emphasise that in the last two decades a marked improvement from within has asserted itself in our science, with constantly increasing success.

The matter has been found simpler than was at first believed. It appears indeed that the different branches of mathematics have actually developed not in opposite, but in parallel directions, that it is possible to combine their results into certain general conceptions. Such a general conception is that of the function, in particular that of the analytical function of the complex variable. Another conception of perhaps the same range is that of the Group, which just now stands in the foreground of mathematical progress. Proceeding from this idea of groups, we learn more and more to coordinate different mathematical sciences. So, for example, geometry and the theory of numbers, which long seemed to represent antagonistic tendencies, no longer form an antithesis, but have come in many ways to appear as different aspects of one and the same theory.

This unifying tendency, originally purely theoretical, comes inevitably to extend to the applications of mathematics in other sciences, and on the other hand is sustained and reinforced in the development and extension of these latter. I assume that detailed examples of this interchange of influence may be not without various interest for the members of this general session, and on this account have selected for brief preliminary mention two of the papers which I have later to present to the mathematical Section.

The first of these papers (from Dr Schönflies) presents a review of the progress of mathematical crystallography. Sohncke, about 1877, treated crystals as aggregates of congruent molecules of any shape whatever, regularly arranged in space. In 1884 Fedorow made further progress by admitting the hypothesis that the molecules might be in part inversely instead of directly congruent. In the light of our modern mathematical developments this problem is one of the theory of groups, and we have thus a convenient starting point for the solution of the entire question. It is simply necessary to enumerate all discontinuous groups which are contained in the so-called chief group of space-transformations. Dr Schönflies has thus treated the subject in a text-book (1891) while in the present paper he discusses the details of the historical development.

In the second place, I will mention a paper which has more immediate interest for astronomers, namely, a resumé by Dr Burkhardt of "The Relations Between Astronomical Problems and the Theory of Linear Differential Equations." This deals with those new methods of computing perturbations, which were brought out first in your country by Newcomb and Hill; in Europe, by Gylden and others. Here the mathematician can be of use, since he is already familiar with linear differential equations and is trained in the deduction of strict proofs; but even the professional mathematician finds here much to be learned. Hill's researches involve indeed -- a fact not yet sufficiently recognised -- a distinct advance upon the current theory of linear differential equations. To be more precise, the interest centres in the representation of the integrals of a differential equation in the vicinity of an essentially singular point. Hill furnishes a practical solution of this problem by the aid of an instrument new to mathematical analysis -- the admissibility of which is, however, confirmed by subsequent writers -- the infinitely extended, but still convergent, determinant.

Speaking, as I do, under the influence of our Göttingen traditions, and dominated somewhat, perhaps, by the great name of Gauss, I may be pardoned if I characterise the tendency that has been outlined in these remarks as a return to the general Gaussian programme. A distinction between the present and the earlier period lies evidently in this: that what was formerly begun by a single master-mind, we now must seek to accomplish by united efforts and cooperation. A movement in this direction was started in France some time since by the powerful influence of Poincaré. For similar purposes we three years ago founded in Germany a mathematical society, and I greet the young society in New York and its Bulletin as being in harmony with our aspirations. But our mathematicians must go further still. They must form international unions, and I trust that this present World's Congress at Chicago will a step in that direction.

**6. Organisation of the Congress.**

By vote of those present it was then resolved to meet in two separate sections, for Mathematics and for Astronomy respectively.

The mathematical section met at 12.30 p.m. in Room 25, where also all its subsequent sessions were held. The assembly was called to order by the chairman of the local committee, Professor E H Moore of Chicago. For the purpose of organization, a nominating committee was chosen, consisting of Professor J M Van Vleck of Wesleyan University, President H T Eddy of Rose Polytechnic Institute, and Professor O Bolza of the University of Chicago. Upon their nomination the following officers were elected unanimously:

President: Professor W E Story of Clark University;

Vice-President: Professor E H Moore of the University of Chicago;

Secretary: Professor H W Tyler of the Massachusetts Institute of Technology;

Executive Committee: the above officers together with Professor Felix Klein of the University of Göttingen, and Professor H S White of Northwestern University.

**6.1. Programme.**

After a short recess the executive committee reported a programme for the week, according to which daily sessions should begin at 9.30 a.m., and the papers and lectures received through the local committee and the commissioner from Germany should be presented as nearly as possible under the following order:

Tuesday, 22 August. Arithmetic, Algebra, Multiple Algebra;

Wednesday, 23 August. Algebraic Curve-Theory, Theory of Functions of a real variable;

Thursday, 24 August. Theory of Functions of a complex variable;

Friday, 25 August. Theory of Groups;

Saturday, 26 August. Geometry.

The committee recommended further that the Congress accept for the afternoons of Tuesday, Wednesday, and Friday the invitation of Professor Klein to visit the German University Exhibit at the World's Columbian Exposition, and attend his exhibition and explanation of mathematical models and apparatus. These recommendations were adopted.

At the session of Tuesday, on motion of Professor E H Moore the Congress by acclamation elected as Honorary President Professor Felix Klein.

Meantime a programme had been printed. The papers at hand being too numerous and extensive for reading in full were given m abstract by their authors if present, otherwise by members designated by the executive committee; or, where this was not possible, were read by title. With this necessary condensation the Congress listened daily to the reading of six papers and the delivery of two lectures, sessions lasting usually three to four hours. On the three afternoons above mentioned, the Congress met at the German mathematical exhibit in the Columbian Exposition at 3 p.m., and attended lectures there given by Professor Klein with the assistance of Professor H Maschke of the University of Chicago.

**7. The final session of the Congress.**

At the final session on Saturday, 26 August, after the regular programme, certain concluding actions were taken. On motion of Professor J M Van Vleck it was voted: That the local committee of the mathematical section of this Congress have authority to make arrangements in regard to the publication of the proceedings and memoirs. On motion of Professor Moore it was voted unanimously: That the thanks of this mathematical section be tendered to Professor Klein for his very valuable contributions to the proceedings of the Congress and for his interesting expositions of the mathematical material in the German University Exhibit at the Exposition.

Remarks were made by Professor A G Webster of Clark University, deprecating the separation, in our educational curricula, of the different branches of mathematical and physical science.

President Story congratulated the section upon the success of their sessions; and in behalf of the section acknowledged its indebtedness to Professor Klein, and the indebtedness of American mathematics in general to the influence and inspiration of German Universities and mathematicians.

**7.1. Adjournment.**

The section then adjourned

*sine die*.