Koenigs henceforth devoted himself to the elaboration of a method of teaching mechanics based on integrating theoretical studies and experimental research with industrial applications. He created a laboratory of theoretical and experimental mechanics designed especially for the study of various types of heat engines and for perfecting different testing procedures. This laboratory, which began operating in new quarters in 1914, played a very important role in World War I.Koenigs was greatly influenced by Darboux and his first work was on geometry following work of Plücker and Klein. He looked at iteration theory in analysis and, following Darboux's plea for a more rigorous approach to analysis, he published papers in 1884 and 1885 on iteration of complex functions using rigorous arguments based on uniform convergence. He showed how to produce eigenfunctions through a process of iteration. He took Ernst Schröder's work on Newton's method, carried out by Schröder in 1870-71, making this theory into a rigorous body of work and in so doing strongly influenced its subsequent development. Koenigs contributions to complex analysis are described in  and these are summarised by Hinkkanen as follows:-
Schröder solved the problem of conjugating an analytic function to a linear map in a neighborhood of an attracting (but not superattracting) fixed point. ... Schröder also discovered an infinite family of generalizations of Newton's method. Schröder's arguments concerning conjugation were not rigorous by modern standards. The work of G Koenigs in 1883-1884 was rigorous and contained, in particular, the results of Schröder. ... Koenigs noted that it would be desirable to determine the partitioning of the plane into domains in which the iterates of a function tend to attracting cycles, but observed that this would be very difficult and perhaps impossible for a general rational function. [He] lacked the concepts of set theory (Cantor sets, fractals, etc.) to be able to study the general situation.Koenigs studied analytic mechanics where he applied Poincaré's theory of integral invariants. In  his work on roulettes is discussed and put into context relating it to the work of several other mathematicians:-
The construction of centres of curvature of plane roulettes by Bobillier (1831), Gilbert (1858) and Koenigs (1897) was based on the theory of centroids by M Chasles (1830). This problem has kinematical origins (a roulette is the plane curve described by points of a plane figure moving in its plane) but each plane curve may be considered as a roulette.From 1910 he worked in his laboratory, concentrating on research in applied thermodynamics. After World War I, Koenigs became involved in the international efforts to oversee cooperation within the scientific community. The International Research Council was set up for this purpose and, during the Constitutive Assembly meeting of the Council in Brussels in 1919, preparations were made to set up the International Mathematical Union. It was decided that an International Congress of Mathematicians should take place in September 1920 and Koenigs, as the French delegate, invited the Congress to Strasbourg. This was accepted unanimously despite the previous decision to hold the next Congress in Stockholm following the 1912 Congress :-
The duly accredited delegates of the (Allied) nations were to meet in Strasbourg at the time of the Congress to confirm the statutes and to create the International Mathematical Union. ... Mittag-Leffler never recognised the Strasbourg Congress as an international event.Koenigs was elected as Secretary General of the Executive Committee of the International Mathematical Union at the Strasbourg Congress :-
Secretary General Koenigs, Professor at the University of Paris (Sorbonne) and Member of the French Academy of Sciences, had not attended any of the pre-war International Congresses. He was elected Secretary General until 1928, but he kept this position for 3 more years until his death in 1931. In light of later events, it is apparent that Koenigs inflicted damage upon the cause of the Union by persisting in maintaining an anti-German policy although times had changed and the passions aroused by the war had largely cooled down. It is curious that Koenigs' archives in the Academy do not contain a single paper associated with the IMU. Nor are the activities of Koenigs in the Union mentioned in the memorial address of the French Academy, in his obituary, or in his biography.In the years after World War I feelings ran high and Germany, Austria-Hungary, Bulgaria, and Turkey were excluded from the Union. This was unfortunate but it was understandable that feelings ran high on the issue at this time. Many, however, felt that mathematics should not be subjected to political pressures but should welcome equally mathematicians of all nations. Leading advocates of that view were Hardy and Mittag-Leffler. Others held equally sincere views that the Union's exclusion rules were right, in particular Koenigs strongly believed in the exclusions. In fact the exclusions were supposed to go further, for only members of the International Research Council were allowed to send delegates to an International Congress of Mathematicians but, although the exclusions of Germany, Austria-Hungary, Bulgaria, and Turkey were adhered to in Toronto, participants from countries which were not members of the International Research Council was allowed. In Koenigs' official report on the Toronto Congress in 1924 he listed the fourteen countries present omitting those that were not members of the International Research Council, then added:
Also present were several scholars from the following countries that have not yet adhered to the Union: Spain, Georgia, Russia, India.Again Koenigs was elected as Secretary General of the Executive Committee of the International Mathematical Union but began to use his position for his own political ends. Pincherle wrote to International Research Council's Vice-President Vito Volterra a year after Toronto. He said he was amazed to learn that the International Research Council had not been informed about his election as President of the International Mathematical Union. He explained that the reason was that Koenigs:-
... frequently and willingly leaves letters unanswered and takes his functions with much calm.The problem concerning who could attend International Congresses became worse for the 1928 Congress when the question concerned the attendance by delegates from Germany. Despite a general softening of views, Koenigs strongly opposed attendance by Germans but first the Executive Committee had to select a location for the 1928 Congress. Bologna and Stockholm emerged as the only competitors. Pincherle, as President of the International Mathematical Union, felt he could not press for Bologna but Koenigs went as far as to say that if Stockholm were chosen the unfavourable exchange rate would stop France and many other countries attending. Bieberbach asked all Germans to boycott Bologna but Hilbert argued that they should attend. However Koenigs, as the International Mathematical Union's General Secretary, argued that Germany should not attend since although they were eligible to join the International Research Council since 1926 they had not done so. He accordingly wrote to all Union members advising them not to take part in the Bologna Congress. Despite his attempts to arrange a boycott there was a large attendance, even from the French. Koenigs himself, however, did not attend.
In January 1929 William Henry Young was elected President of the International Mathematical Union. Koenigs was still General Secretary but was experiencing health problems and was not responding to any letter sent to him. Henry Lyons, Secretary General of the International Research Council, wrote to William Henry Young saying that:-
... as Koenigs has never replied to any of my previous letters, he is not likely to do so to this.Young was advised by de la Vallée Poussin to replace Koenigs, with Fehr being suggested as his replacement as General Secretary. Young did nothing, however, so in early 1931 de la Vallée Poussin informed everyone in the Bureau about his proposal to replace Koenigs. The matter was still unresolved when Koenigs died in October 1931. Lehto writes :-
The presence of Koenigs in the Bureau not only meant that secretarial work was blocked, but as a symbol of the discrimination policy now condemned, it cast a shadow over the Union. Despite Young's attempts to save the IMU, it was suspended in 1932.Let us mention some of Koenigs most important works: Mémoire sur les lignes géodésiques Ⓣ (1892), La géométrie réglée et ses applications Ⓣ (1895), Leçons de Cinematique: Professees a la Sorbonne Ⓣ (1895-97); and Mémoires sur les courbes conjuguées dans le mouvement relatif le plus général de deux corps solides Ⓣ (1910). Perhaps Leçons de Cinematique Ⓣ is the most famous of all. Lovett , reviewing the book, writes:-
Koenigs' treatise promises to be a classic. In facing the formidable array of researches and methods of Poinsot, Chasles, Bonnet, Ribaucour, Darboux, and a host of others the author must have experienced no little difficulty in choosing a method of exposition of the subject matter. A treatise that is to be both scientific and didactic must consider the demands both of the student and of the scientist. Knowledge is of most use when accompanied by a kit of tools, and the instructor's art is a double one - he must not only present the facts of a subject with reasons for faith in them, but also employ those methods which promise results at the hands of the independent investigator. Further, the method should have the necessary breadth and unity, and be possessed of the clearness and directness of geometry and the power and generality of analysis. This multiplicity of demands Professor Koenigs has met admirably by basing his exposition on the geometry of the straight line and employing the mobile trieder of reference; the latter in the hands of Ribaucour and Darboux has proved itself to be the most certain and powerful implement yet used in infinitesimal geometry and it naturally lends itself with equal facility and elegance to the geometry of displacement.In addition to these books, Koenigs published around sixty papers.
He was honoured for his achievements in a number of ways. The Bordin Prize of 1890 was to be awarded for work on differential geometry; Koenigs won for his essay on geodesics. He also received the Poncelet Prize in 1913 for his contributions to geometry and mechanics. He was elected president of the French Mathematical Society in 1897 and of the French Society for Aviation Navigation in 1914. He was elected to Mechanics section of the Academy of Sciences on 18 March 1918, having failed to be elected in 1892.
Article by: J J O'Connor and E F Robertson
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