**Eustachy Żyliński**was born in Kuna, a village about 5 km south west of Haisyn (or Gaysin as it was known at the time of his birth) in the Russian Empire. He was the only child of Janusz Mikolaj Żyliński (known as Jan) and Kazimiera Zmudzinskich. Jan worked as a clerk but the family were quite poor. When he was ten years old, Eustachy entered the First Gymnasium in Kiev where he studied for the next ten years. He was an outstanding pupil and, on 7 June 1907, he graduated from the Gymnasium with a gold medal. He then entered the Imperial University of Saint Volodymyr in Kiev where he studied in the Department of Physics and Mathematics. He graduated with a first degree from the University of Kiev in 1911 after submitting his dissertation

*Foundations of the theory of ordered rational numbers*(Russian). He was awarded a silver medal for outstanding work for the essay

*Recent developments in the theory of ideals*(Russian), submitted as a competition entry to the Department of Physics and Mathematics. He had been taught at the University of Kiev by Dmitry Aleksandrovich Grave who published

*Elementary course in number theory*(1909-10). In his memoirs, Grave acknowledges Żyliński's help with this first edition of his book.

In 1912 Żyliński was awarded a 4-year scholarship to fund his studies abroad so that he could undertake research with the aim of becoming a professor of mathematics. He spent the year 1912-13 at the University of Göttingen, in Germany, studying under Edmund Landau. He published *Zur Theorie der ausserwesentlichen Diskriminantenteiler algebraischer Körper* Ⓣ in *Mathematische Annalen* in 1913, submitted in July 1912 while he was visiting Göttingen. In this paper he extended results published earlier by Kurt Hensel. He then went to the University of Marburg where he studied with Hensel and finally he spent time at the University of Cambridge in England where he studied with G H Hardy. In 1914 he was awarded his Master's Degree (equivalent to a Ph.D.) by the University of Saint Volodymyr in Kiev after taking examinations in algebra and number theory, and defending his thesis *On the domain of p-adic numbers*. In addition to his foreign visits, Żyliński participated in Grave's seminar in Kiev during the period 1912-1916. In April 1916 he was drafted into the Russian army and, for the first few months, completed his engineering degree at the Engineering School in Kiev. In September 1916 he was sent to St Petersburg where he attended the Electrotechnical University. He took various courses there, passing examinations on the internal combustion engine on 6 October 1916, on electrical engineering on 22 November 1916, and on telegraphy on 6 February 1917. He was promoted to second lieutenant at the time of unrest which began in St Petersburg near the end of February and led eventually to the Russian Revolution. He served in Berdichev and Kamianets-Podilskyi, lecturing on electrical topics, at the time when the Russian Empire was collapsing and Ukraine was seeking independence.

Żyliński's military service ended in November 1917 and, shortly afterwards, he was appointed as an assistant professor of mathematics at the Polish College of the University of Kiev. This was the beginning of a short period of independence for the Ukraine with Kiev as the capital of the Ukrainian People's Republic. In addition to teaching analytic geometry, set theory and higher algebra at the Polish College, he also taught at the Ukrainian State University and at the Higher Technical Institute. While in Kiev he had volunteered to serve in the 1^{st} Polish Corps and, in February 1919, he went to Warsaw where he served for a few months as an officer in the Polish army. His duties were as a lecturer teaching electrical topics at the Military School. He was offered a position at the Ukrainian State University in Kamianets-Podilskyi but turned this down in favour of a position at the Jan Kazimierz University of Lwów (now Lviv in Ukraine). After 120 years of partition, Poland had become an independent country following World War I and the Polish government had named the University after Jan Kazimierz in November 1919. It was the third largest Polish university after Warsaw and Kraków. Żyliński took up his appointment as associate professor of mathematics at the University of Lwów on 1 October, having been released from the army in the previous month at the request of the University of Lwów.

The position in Lwów became vacant following the death of Józefa Puzyny (1856-1919) and the departure of Wacław Sierpiński following his move from Lwów to Warsaw. Żyliński became a colleague of Hugo Steinhaus and soon they were joined in Lwów by Stefan Banach who was an assistant at the Lwów Technical University from 1920. Banach habilitated at the Jan Kazimierz University of Lwów in 1922. On 27 July of that year Żyliński was appointed as a full professor of mathematics at the Jan Kazimierz University. Herman Auerbach (1901-1942) was appointed as his assistant in 1923. During these years in Lwów, he had been living with his mother. However, in 1925 he made a home of his own when he married Kazimiera Ramlau (1900-1976) on 16 April. Kazimiera's parents, Aleksander Ramlau and Zofia Kepinska. had both died in 1904 when she was only three years old.** **Eustachy and Kazimiera had two children, a daughter Maria (1926-1990) and a son Janusz (1931-).

Mark Kac, who began his studies of mathematics in Lwów in September 1931, describes Żyliński as a lecturer [1]:-

Kac also describes a rather unusual mechanics examination by Żyliński [1]:-The first lecture, which I attended was in theoretical arithmetic. This course was designed to introduce in rigorous, axiomatic way the number system. It started with the famed axioms of Giuseppe Peano for non-negative integers and ended with irrational numbers. This was my first encounter with abstract mathematics and I took to it like a duck to water. In fact I was so good at it that the professor who taught the course, Eustachy Żyliński, asked me to help in preparing lecture notes which were mimeographed and, I believe, sold. I was so pleased to have been thus singled out that I failed to notice that no remuneration was offered for what had been a considerable amount of work. My name, however, did appear on the title page, though less prominently than the name of the professor, who had done almost no work at all. Concurrently, I attended a proseminar on algebra and number theory. It was very loosely connected to the course in theoretical arithmetic and it was under the nominal supervision of Żyliński. It was run, however, by Marcel Stark, a junior assistant at the Mathematical Institute.

However, Kac makes a rather unfriendly comment when explaining that he needed support for an application for a fellowship. He wrote:-It is curious that I drew Żyliński as an examiner four out of five times. The a priori probability of this happening is a little less than1.5percent, small enough to cause one to raise one's statistical eyebrows. Żyliński was very polite to me, and when I arrived at his office for the oral examination in mechanics, he looked up at me from some work, which he pored over, and said, "Oh, it's you. You have an A." When I said: "But won't you ask me at least one question?" He said, "Very well. Can you tell me what is the Schrödinger equation?" I said: "I don't know," and then he said: "All right. You still have an A". The Schrödinger equation is the fundamental equation of(non-relativistic)quantum mechanics, which was not part of the syllabus, and I am quite sure that Żyliński himself didn't know the Schrödinger equation.

World War II meant that the University of Lwów, and all its staff, went through incredibly difficult times. At the start of World War II in 1939, Russia and Germany had a pact, the so-called Ribbentrop-Molotov pact, to divide Poland between them. The two-pronged attack, the Germans from the west and the Russians from the east, quickly defeated the Polish army and Lwów was taken over by the Soviet Union. Although it was now under Soviet control, the university attempted to continue to operate as normally as possible despite many arrests and deportations carried out by the occupiers. This was not easy for, in the period from September 1939 to June 1941, the Soviets murdered over a dozen of the Jan Kazimierz University faculty. Żyliński was called up as an army reservist in the summer of 1939 and at that time his wife, together with their children, left Lwów and went to live in her home town of Szczytniki; they did not return to Lwów until 1943. Żyliński was in Lwów when the Soviets attacked in September 1939. He became head of the Department of Algebra in December 1939 and survived with difficulty in this role until June 1941. At this time everything changed for in June 1941 Germany attacked its former ally, the Soviet Union. On 1 July 1941 German troops entered Lwów and on the following day 36 professors who had worked at the university were arrested. All but one of the professors were shot by the Germans; the university could no longer operate and was closed. Żyliński worked as a statistician in the transport office during the German occupation but, although the Jan Kazimierz University of Lwów was officially closed, secret teaching was organised by a number of the former staff. Władysław Orlicz and Żyliński were the main mathematics teachers at this underground university. In August 1944, the Soviets took control of Lwów again and, at this time, they opened the State University of Lwów with Żyliński as head of the Department of Algebra and Number Theory.To have a chance at all, I needed a strong recommendation. Stark wouldn't do because he was too junior, and I didn't quite trust Żyliński.

As the war entered its final stages, it was clear that Lwów would not remain part of Poland when treaties were drawn up. In August 1945 it was announced that Lwów would become part of the USSR and Żyliński, together with Stanisław Mazur who was also teaching in Lwów, signed up for voluntary repatriation to Poland. In May 1946 Żyliński, together with his wife and children, left Lwów and travelled to Łódź. He worked for the Ministry of Foreign Affairs for a while, being appointed as consul to Kiev, but he resigned this position. From 1 October 1946 until he retired on 30 September 1951 Żyliński was a professor and head of the Department of Mathematics, Engineering and Construction at the Silesian University of Technology in Gliwice. He lived at the University of Gliwice while working there, but retained a permanent home in Łódź. After he retired he continued to live in Łódź but seems to have given up mathematics. His health deteriorated and he suffered a series of strokes. He died in the Barlickiego Hospital after suffering his third stroke. He was buried in the St Joseph Roman Catholic Cemetery in Łódź. His wife Kazimiera continued to live in Łódź with their daughter Maria who did not marry. Kazimiera died in 1963 and was buried beside her husband.

Concerning Żyliński's mathematical contributions, Lech Maligranda writes in [3]:-

These textbooks are (all in Polish): (with Stanisław Ruziewicz)Eustachy Żyliński worked in number theory, algebra, logic and foundations of mathematics. He has written over20scientific papers and6textbooks.

*Algebra: handbook for the higher classes of secondary schools*(1926); (with Stanisław Ruziewicz)

*Algebra: handbook for the higher classes of middle schools*(1926); (with Stanisław Ruziewicz)

*Introduction to Mathematics, Algebra*I (1927); (with Stanisław Ruziewicz)

*Algebra: handbook for the higher classes of secondary schools*(1928);

*Introduction to arithmetic theory*(1932); and

*Analytical Geometry*(1938).

Marcelli Stark entered the Jan Kazimierz University of Lwów as an undergraduate in 1926. He later became an assistant at the university and a colleague of Żyliński. He was much influenced by Żyliński's courses on Analytical Geometry, assisting in their teaching, and published *Analytic Geometry with Particular Regard to the Textbook of Eustachy Żyliński* in 1951. Further editions of this popular text appeared in 1958, 1967, 1970 (enlarged), 1972 and 1974. Vaclav Hlavaty writes in a review of the 1951 edition:-

Finally let us note Żyliński's interests outside mathematics. He enjoyed hunting, played tennis, skied and was also a bridge player.This textbook deals with metric, affine and projective geometry of linear and quadratic varieties in the plane as well as in the three-space. The author does not confine himself only to real plane(real space)but considers also the complex plane(complex space). Besides items usually dealt with in textbooks of elementary analytic geometry the reader finds here the introduction to synthetic projective geometry, to the theory of matrices(and determinants)with the usual applications and to the(three-dimensional)elementary vector calculus.

**Article by:** *J J O'Connor* and *E F Robertson*