The course that König followed in the Medical School of Vienna University included a thorough study of mathematics and the natural sciences. Feeling now that he wanted to follow a course in the natural sciences, although still retaining his medical interests, he went to Heidelberg which had an excellent reputation for science with famous men such as Bunsen, Kirchhoff and Helmholtz on the teaching staff. His first research paper was written on a topic suggested by Helmholtz and considered a medical topic. The paper was published by the Vienna Academy of Sciences in 1870.
Perhaps the most significant event which turned König's interests firmly towards mathematics happened in 1869 when Königsberger left Greifswald to take up the appointment of a chair of mathematics at Heidelberg. Königsberger had been greatly influenced by Weierstrass's lectures on elliptic functions and this was the topic which interested him at this time, so he in turn influenced König to also undertake research on elliptic functions. König worked on his doctoral dissertation under Königsberger's supervision and submitted his thesis Zur Theorie der Modulargleichungen der elliptischen Functionen Ⓣ to Heidelberg in 1870. He was awarded his doctorate in June 1870, and his thesis was published in the following year.
After being awarded a doctorate from Heidelberg König went to Berlin where he spent six months attending lectures by Weierstrass and Kronecker. He then returned to Budapest where he was appointed as a dozent at the University in 1871. He became a professor at the Teacher's College in Budapest in 1873 and, in the following year, was appointed professor at the Technical University of Budapest. This was a period when improvements in Hungary's economic position led to a need for education and consequently a need to train more school teachers. The mathematical level began to rise steadily and the Technical University of Budapest was established in 1871 with the right of issuing diplomas. König joined the University at a time when there were other talented and enthusiastic mathematicians on the staff. König took a lead in a new venture which these mathematicians embarked on, namely the founding of a new mathematical journal Müegyetemi Lapok. The first issue appeared in 1876 and a part was produced each month. Sadly only 3 volumes and 30 issues appeared before it was forced to close. The editors wrote in the final issue:-
With this issue the Technical University Journal has come to the end of its career. It seems that here in Hungary no mathematical journal can exist without financial support. Of course, if there were only half as many readers of mathematics as there are teachers, things would be quite different.It was with great sadness that König was forced to admit that the venture which he had thought so highly of could not continue after so short a time.
Perhaps it was to teaching, rather than to research, that König made the most important contributions. He raised the level of mathematics teaching at the Technical University to a high standard, in fact it was a level which the engineers must have found very demanding. Szénássy writes :-
König was extremely conscientious and he expected the same of his students. He went into details whatever the subject be and analysed everything thoroughly even for students of the Technical University. He emphasised the importance of the quality rather than the quantity of knowledge, and he tried to get his students to acquire an exact mathematical thinking.He contributed to the Technical University in other roles too. He was on three occasions Dean of the Engineering Faculty and also on three occasions he was Rector of the University.
König worked on a wide range of topics in algebra, number theory, geometry, set theory, and analysis. One of his early ideas was a paper of 1872 which looked at intuitive ways to prove the consistency of non-Euclidean geometries. He published many research papers in analysis, but his greatest significance in this area comes from the excellent textbooks which he wrote on the topic. His most important work written in 1903 is based on a fundamental study by Kronecker published in 1892. König developed Kronecker's polynomial ideals and presented many results on discriminants of forms, elimination theory and Diophantine problems. König's work on polynomial ideals influenced Hilbert, Lasker, Macaulay, Emmy Noether, van der Waerden and Gröbner but they simplified his ideas so König's work is now only of historical interest.
In the last eight years of his life König's interests turned towards set theory and he contributed to the continuum hypothesis. In August 1904, at the International Congress of Mathematicians at Heidelberg, he announced that the continuum hypothesis was false :-
... what a sensation the announcement of the title of König's lecture .. stirred among the participants of the Congress. All section meetings were cancelled so that everyone could hear his contribution.König's proof contains an error in that he applied a theorem due to Felix Bernstein in a case where it does not hold. It was a little while before Zermelo found the error in the proof and then in 1905 Felix Bernstein published a short note correcting his theorem. König retired from his post at the Technical University in 1905 but he continued to lecture there particularly on topics that he was interested in. Clearly retirement had been undertaken so that he could spend more time on things which he wanted to do - is this not the reason why many academics take early retirement? He spent the last part of his life working on his own approach to set theory, logic and arithmetic, which was published in 1914, the year after his death. He had been working on the final chapter at the time of his death.
We should mention König's contribution to mathematics teaching at secondary schools in Hungary. He worked out the algebra part of the syllabus and wrote some fine textbooks to support the teaching. He also contributed to the Hungarian Academy of Sciences, which he was elected to in 1889, serving on the Board for 19 years. As a member of the Board of Directors of the largest publishing house in Hungary he had a wealth of expertise and experience to share with his colleagues.
König married and had two sons; Dénes was a talented mathematician, while György was an expert on the history of literature. We mentioned at the beginning of this article that in his young days König had a love of literature, so it is easy to see why one of his sons might have followed this career. König retained his love of literature and reading was throughout his life his favourite hobby.
Article by: J J O'Connor and E F Robertson