VLADIMIR VARIĆAK

(1865 - 1942)

Vladimir Varićak was born on March 1,1865, in a village Švica near Otočac in Croatia, in a Serbian family. He attended elementary and secondary school in Sisak, Petrinja and Zagreb. In the period 1883-87 he studied mathematics and physics at the Faculty of philosophy, University of Zagreb. He became an advanced university student in 1887, and passed professorial exam in mathematics and physics in 1888. In 1891 he finished his Ph.D. thesis, and in 1995 he obtained the habilitation thesis for the position of private docent. He spent the first ten years of employment working in secondary schools in Zemun, Zagreb and Osijek, and in navigation school in Bakar. In October 1889 he became professor of mathematics at the Faculty of philosophy, Zagreb, and remained at that position until 1936, when he retired. After that he continued to work as a part-time professor at the same faculty, where he spent 42 years as a lecturer, until the end of his life (January 17,1942).

Vladimir Varićak was the member of the following institutions: Yugoslav academy of sciences and arts; Czech academy of sciences; Serbian academy of sciences; Croatian society for natural sciences and Yugoslav mathematical society. In the year 1925 Varićak took part in the competition for the Lobachevsky prize, and received a diploma of the Physical-mathematical Faculty of Kazan for his contributions to Lobachev-skian geometry and its application to Einstein's theory of special relativity. Varićak was creatively engaged in teaching of mathematics at the level of both secondary school and university. He had a great influence on our two well-known scientists: on Milutin Milanković, as his teacher at the secondary school in Osijek, and on Đuro Kurepa, as the university professor. In his memoirs Milanković wrote: Among all the teachers, Varićak had the greatest influence on me. And Kurepa, the most prominent student and collaborator of Varićak, described his lectures in the following way: In a very small number of universities in the world, at that time, one had the opportunity to learn about so many things, and all that in lectures given by a single person.

The scientific work of Vladimir Varićak was devoted to the investigation of Lobachevskian geometry and special relativity, and he also studied life and work of Ruđer Bošković. In the first period of his studies of Lobachevskian geometry Varićak relied on the Poincare model. Later, he suggested and developed a specific approach to this subject. He derived the equations of straight lines and planes, equidistant lines and surfaces, limiting circles and limiting spheres. He also studied various transformations in Lobachevskian plane. His work was characterized by a clear geometrical idea, which was first realized on some conveniently chosen specific cases, and then generalized. After that, by analyzing the general case, Varićak was able to find a lot of new and interesting special cases. In 1907 Varićak published a paper First founders ofnon--Euclidean geometry, where he gave not only a historical review of the development that led to the discovery of nonEuclidean geometry, but also analyzed the consequences of that discovery, with a very detailed and convincing discussion of important and critical moments of that development.

As an expert in non-Euclidean geometry, Varićak was able to recognize, very soon after the discovery of special relativity (SR), a specific connection between the new mechanics and non-Euclidean geometry. When Sommerfeld, by the end of 1909, interpreted SR with the help of geometry on a pseudosphere, Varićak clearly understood that this unusual geometry is nothing else bat a representation of Lobachevskian geometry. In the next two years he published five papers on the non-Euclidean interpretation of the new theory. These papers represent a basis of his contribution to the non-Euclidean interpretation of SR, a subject he continued to work on for the next nearly thirty years. Varićaks work was well known to his contemporaries.

In his studies of SR Varićak tried to replace Euclidean geometry of the classical mechanics by three-dimensional Lobatschevskian geometry, and was able to obtain in that way many important results of SR. On the other hand, it was clear very early that the geometrical structure of SR is best described by fourdimensional Minkowskian geometry. A critical analysis of Varićak's ideas shows that the structure of three-dimensional Lobatschevskian space is not sufficient to describe all the events of four-dimensional Minkowskian space, while the use of Lobatschevskian geometry to describe the space of velocities is completely correct. Although in many cases the two interpretations lead to the same results, they are not mathematically equivalent. The main contribution of Varićak's work lies in his understanding that SR demands a non-Euclidean interpretation, which was very difficult to accept in the early days of this theory, and also in his attempts to develop a systematic, non- Euclidean interpretation of SR, in spite of the fact that these attempts were only partially successful.

Varićak investigated life and work of Ruđer Bošković with a great interest. He studied historical documents about Bošković in Milan, Rome, Vienna and other European towns, and published about twenty papers on that subject. From Varićak's papers about Bošković one can learn a lot about Varićak himself. In his studies of Bošković's mathematical work Varićak used a specific methodology that was very appropriate for the history of mathematics. His investigations of Bošković's life and work were recognized by the international community.

Authors: Prvanović, Mileva ; Blagojević, Milutin