**András Rapcsák**'s parents were Péter Rapcsák, who was an inn keeper, and Mária Bereczki who ran a small grocer's shop. András had two brothers: Péter Rapcsák who was three years older than András and became a Calvinist priest; and István Rapcsák who became a radiologist. András attended both primary and secondary schools in his home town of Hódmezövásárhely which is about 20 km north east of Szeged. Between 1925 and 1933 he attended the Bethlen Gábor Calvinist Gymnasium in Hódmezövásárhely. This school had a long tradition having been founded in 1723 but it was only given the name Bethlen Gábor in 1930, during the time that Rapcsák studied there. In fact 1930 was the year that Rapcsák's brother Péter graduated from the same school. In 1993 Rapcsák recalled his school years (see [1]):-

At this school Rapcsák was in the same class as Márton Sain (1915-1997) who went on to become a well-known historian of mathematics and a gymnasium teacher. Although Rapcsák did not excel at school, in his final year at the Bethlen Gábor Calvinist Gymnasium he took part in a national mathematics competition. This did result in him learning about the great Hungarian mathematicians Farkas Bolyai and his son Janos Bolyai. Learning about non-Euclidean geometry made Rapcsák think deeply about the nature of space and he gave a talk on this topic to the Literary Society of the Gymnasium.I started secondary school at the age of eleven, and I went there for eight long years. When I first entered through its immense gate with oak panels, I immediately caught sight of the inscription carved in the floor: "Blessing on all the stones of this house." My teachers taught me everything in this spirit. It was here that I learnt how to hold out in life and to complete my duties. I learnt here that I must devote my life to beauty and truth and that I must live with dignity. Furthermore, my passion for mathematics and poetry also began here.

Graduating from the Bethlen Gábor Calvinist Gymnasium in 1933, later that year he entered the Ferenc József University of Szeged taking courses in mathematics and physics. He was taught by several famous mathematicians such as Frigyes Riesz, Alfréd Haar, László Kalmár, Béla Kerékjártó, and István Lipka (1899-1990). His mathematics tutor was László Kalmár. However, he continued his interest in poetry and he joined the Szeged Society of Young Artists which at this time had fifteen members. There he became friends with the poet Miklós Radnóti (1909-1944) who was studying for a doctorate in the Humanities Faculty. Radnóti was Jewish and died in the Holocaust ten years later. In the Art Society, Rapcsák also became friends with Gábor Tolnai (1910-1990), an art historian who had graduated from the University of Szeged in 1933.

Rapcsák showed his talents for mathematics, winning prizes for two of his essays, but his undergraduate years were made exceptionally difficult due to a serious illness which was diagnosed when he was nineteen years old. This illness was osteosarcoma, cancer of the bones, which most commonly affects the leg bones of teenagers. In Rapcsák's case he suffered constant pain in his legs and the cancer was diagnosed. He was treated using radiotherapy which halted the progression of the cancer. This, however, was not a cure and for many years he could only walk with great difficulty using two sticks. This illness caused him to miss three years of his university studies. But this was not the only problem he faced for, by this time, his parents were in financial difficulties and so, in parallel with his studies, he worked as a tutor in a student hostel in Hódmezövásárhely from 1937 to 1942 to help support his parents. Other events in Europe were making life difficult too. World War II began in 1939, in November 1940 Hungary signed a pact with Germany and by 1941 it was involved in military action. During these years Rapcsák was completing him university studies and he graduated in 1942, nine years after his first matriculation.

After graduating, Rapcsák began a career as a school teacher. His first position was at the Lutheran School in Rozsnyó (now Roznava in Slovakia). He was only there for a short time but it proved very important for him since at that school he met Jolán Baranyai (born 1911), a teacher of geography and history, whom he later married. They had three children: András Rapcsák, born 14 July 1943 in Debrecen; Tamás Rapcsák, born 18 March 1947 in Debrecen; and Marianna Rapcsák, born 1948. Let us give some details of these children before continuing our biography. András's school education was in in Debrecen, he was an undergraduate in Budapest and took a higher degree in electrical engineering in Debrecen. After working as an electrical engineer, he became a politician. He died in 2002. Tamás became a mathematician and worked on operational research. He became Head of the Department of Operations Research and Decision Systems in the Computer and Automation Research Institute of the Hungarian Academy of Sciences. He died on 24 March 2008. Marianna became a biologist and worked in the Department of Physiology of the University of Debrecen. She died in 2014.

Let us return now to the subject of this biography. After the short spell in Rozsnyó, Rapcsák taught in the Calvinist Reformed Gymnasium in Debrecen which prepared pupils for college, in the Unified Boys' Gymnasium in Debrecen, as well as in two Primary Schools in Debrecen. Teaching started in the Unified Boys' Gymnasium in December 1944 in the buildings of the Reformed School after the Soviet army had left. It was not possible to open the five Gymnasiums in Debrecen due to war damage so this one unified school was opened. In [1] his contributions as a teacher are discussed:-

However, in addition to his school teaching, from 1945 Rapcsák also taught at the University of Debrecen and at the Teacher Training College. The professor in the Institute of Mathematics was Ottó Varga (1909-1969) who founded the Hungarian school of differential geometry. His assistant at the time was Béla Gyires. It was coming in contact with Varga that changed Rapcsák's career for Varga saw that he had great mathematical talents and persuaded him to undertake research for a doctorate. Now Varga's main interests were in differential geometry so it was natural that he should direct Rapcsák in that direction. Continuing with his teaching career, Rapcsák undertook research on differential geometry advised by Varga and submitted his thesisHe was a real scholar teacher. As a secondary school teacher he undertook didactics research work, and he wrote schoolbooks for the students of technical secondary schools with coauthors(L Gyarmathi, I Csánk, S Török). These books were very popular and they ran to10-19editions. The figures were drawn by his fellow teacher J Erdösi. With his college L Magyari he also wrote a schoolbook, but it could not appear in printed form at the time of World War II. He wrote a supplement for the students of the technical secondary schools entitled 'Complex Numbers'(1953). He was open minded to problems of teaching mathematics. In his article 'The treatment of geometry in secondary schools' he investigated the problem of teaching geometry by the help of axioms.

*The theory of surfaces in Minkowski space*(Hungarian) in 1947. In 1949 he published

*Kurven auf Hyperflächen im Finslerschen Raume*Ⓣ which was described as follows in a review by Akitsugu Kawaguchi:-

The Eszterházy Károly Lyceum in Eger was founded in 1774 but the government decision to have only one university in Hungary which was made three years later meant that the Lyceum could not teach university level courses. It continued as an educational establishment and was the Archdiocesan Teachers' Training College from 1852 to 1948. The Hungarian Parliament transferred the Pedagogical Institute from Debrecen to Eger in 1948 and, from 1949, the Lyceum became the Eger Teachers' Training College. Now Rapcsák had taught at the Pedagogical Institute in Debrecen and, when it was transferred to Eger he was made Head of the Department of Mathematics there in 1949. From 1949 to 1951 Rapcsák taught in Eger but it was no easy role he had to fill for, in his own words:-Consider an arbitrary curve on a hypersurface in an n-dimensional Finsler space; then the curve has n-1invariants(i.e., curvatures)as a space curve but n-2invariants as a curve on the hypersurface. The paper finds the relations between these systems of invariants. These relations contain formulas which can be regarded as generalization of the Darboux relations for the motion of the moving trihedra of a surface. It should be remarked that the author uses as the absolute differential on the hypersurface the one induced orthogonally from the absolute differential in the space, following Ottó Varga[1941].

However, he achieved this multi-role job with great success. Béla Pelle, who was a student at Eger and went on to become a professor there said (see [1]):-I had to be a professor, an associate professor, a senior lecturer and an assistant lecturer all in one person.

Rapcsák himself spoke about his time at the Lyceum in Eger:-A few of us in Eger did our Ph.D. under Rapcsák's supervision. He was one of the reviewers of my geometry textbook. He played an important role in saving the Department of Physics in Eger. He talked to the government official until he succeeded in persuading him not to close the Department. He also gave demonstration classes at the teacher training school. When he entered the classroom, he already knew the names of every one of the pupils. His classes were marvellous.

Today the Lyceum in Eger is the Eszterházy Károly University of Applied Sciences and, in addition to the 18I emphasized that success in mathematics depends upon understanding things. Cramming formulae results in a hate of mathematics, because students are drowning in the flood of data, and without a lifebelt they will never reach the shore of spirit. A teacher must have a profound knowledge, and instead of brooding on what he should teach, he should rather concentrate on how he can reach his aim. Students are the most terrible, but at the same time the fairest judges as well. A teacher cannot conceal his ignorance before them. On the other hand, young people esteem openness and knowledge very much. Do not try to deceive them, otherwise you will be completely discredited. Young minds must always be polished with humanity.

^{th}century buildings in the centre of the city (which includes a camera obscura dating from 1776) there are now new buildings on the edge of the city where the mathematics department is housed.

In 1951 Rapcsák returned to Debrecen when he was appointed as an associate professor at the University of Debrecen. He became a candidate of science in 1955 and, in the same year published two papers: (i) *Theorie der Bahnen in Linienelementmannigfaltigkeiten und eine Verallgemeinerung ihrer affinen Theorie* Ⓣ which generalises the geometry of paths as developed by Jesse Douglas in 1928; and (ii) *Invariante Taylorsche Reihe in einem Finslerschen Raum* Ⓣ which was reviewed by Louis Auslander:-

In Hanno Rund's bookHarold S Ruse(1931)derives an invariant Taylor expansion for tensors on a Riemann manifold. His method depends essentially on the use of normal coordinates. Now Herbert Busemann(1950)has shown that the normal coordinates of Riemannian geometry are not in general valid in a Finsler space. The author, however, using the new normal coordinates introduced by Ottó Varga(1950,1952)succeeds in generalizing the calculations of Harold S Ruse to Finsler space.

*The Differential Geometry of Finsler Spaces*(1959) there are references to results by Rapcsák contained in the following of his papers:

*Kurven auf Hyperflächen im Finslerschen Raume*Ⓣ (1949);

*Invariante Taylorsche Reihe in einem Finslerschen Raum*Ⓣ (1955);

*Eine neue Definition der Normalkoordinaten im Finslerschen Raum*Ⓣ (Hungarian) (1954);

*Eine neue Charakterisierung Finslerscher Raume skalarer und konstanter Krümmung, und projektiv-ebene Raume*Ⓣ (1957). In this last mentioned paper, Rapcsák generalises well-known theorems on Riemannian spaces of constant curvature to Finsler spaces. Hanno Rund also reviewed Rapcsák's paper

*Metrische Charakterisierung der Finslerschen Räume mit verschwindender projektiver Krümmung*Ⓣ (1957) writing:-

Rapcsák had a great love for teaching his students and put much effort into preparing his lectures. He liked to experiment with different ways of teaching a topic, especially the differential and integral calculus, to see which would be the most effective.Finsler spaces of constant and of scalar curvature were defined by Berwald[(1947)], who also introduced the notion of spaces of vanishing projective curvature, the latter being more general than the former. In the present paper the author introduces certain hypersurfaces by means of which the above-mentioned special types of Finsler spaces may be characterised. In the treatment of these hypersurfaces the element of support is taken in the direction of the unique transversal to the hypersurfaces. Hyperplanes are defined by the condition that the unit transversal vectors should be parallel(with respect to the metric of the imbedding space). Necessary and sufficient conditions for the existence of such hyperplanes are found; these conditions may then be used to distinguish between spaces of zero projective curvature and spaces of constant curvature. Conditions that the metric of the hyperplanes be Riemannian and of constant curvature are also derived.

In 1960 Rapcsák was awarded a D.Sc. and was made a full professor in the following year. He headed the Department of Analysis at Debrecen University from 1965 to 1968. From 1977 to 1980 he was the Director of the Institute of Mathematics and, from 1984 to 1985, he headed the Department of Probability and Applied Mathematics. He also took on roles such as vice dean of the Faculty of Sciences (1954-1955), vice rector of the university (1955-1957 and 1959-1963), dean of the Faculty of Sciences (1965-1966), and he was the rector of the university between 1966 and 1973.

The illness which he had suffered as an undergraduate had never been cured, the treatment preventing its further development. Certainly the treatment was effective since Rapcsák lived to his late 70s, but eventually he died as a result of a recurrence of the cancer.

He received the following honours [1]:-

Silver grade of the Medal of People's Republic of Hungary(1952), Eminent Pedagogue(1955), Silver grade of the Order of Labour(1965), Honorary doctor of Taras Shevchenko University of Kiev(1968), Golden grade of the Sport Medal(1969), Golden grade of the Order of Labour(1974), The freedom of Debrecen city(1974), Commemorative plaque on the25(1975)^{th}anniversary of the Faculty of Sciences, Pro Universitate Award(1977), Order 'for the Socialist Hungary'(1984), Commemorative plaque(1985), Doctor honoris causa of Lajos Kossuth University(1988).

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