# Ágoston Scholtz

### Born: 27 July 1844 in Kotterbach, Zips district, Austro-Hungary (now Rudnany, Slovakia)

Died: 6 May 1916 in Veszprém, Hungary

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**Ágoston Scholtz**was born in the village of Kotterbach in the Szepes district of the Austro-Hungarian Empire. This region remained part of the Austro-Hungarian Empire throughout Scholtz's lifetime but in 1920, a couple of years after his death, the Treaty of Trianon divided the Austro-Hungarian Empire and this district became part of Czechoslovakia. In 1992 the district became part of Slovakia. A consequence of these changes is that the names of the towns have changed and we will give the Hungarian names, as they were when Scholtz lived there, as well as the present Slovak names. We also try to give the German names of the towns but the situation can be quite complicated. For example Kotterbach, the town where Scholtz was born, was known as Otosbanya or Kotterbach in Hungarian, Koterbach or Koterbachy in German, and today is Rudnany in Slovakia. Kotterbach is a small village near the city of Igló, the biggest town in the district, which also has a collection of different names such as Neuendorf or Neudorf in German, Igló in Hungarian, and its present name of Spisska Nova Ves in Slovakia. The district in which these towns are situated was called the Szepes district in Hungarian, the Zips district in German, and is today the Spis district of Slovakia. Kotterbach owes its existence to the mining industry, being the place where workers in the silver, iron, mercury and copper mines lived. Given this, it will come as no surprise to learn that Scholtz' father was a supervisor of mines.

Scholtz attended grammar school in the towns of Igló, Rozsnyó (Rosenau in German and today Roznava in Slovakia) and Löcse (Leutschau in German and today Levoca in Slovakia). After completing his secondary education he studied at the universities of Vienna and Berlin, where he was awarded a degree in 1865. His first employment was teaching mathematics at the grammar school of Igló, where his younger brother was studying so he his own brother's mathematics teacher.

From 1871 he was a teacher of mathematics and natural philosophy at the Lutheranian Grammar School of Budapest which at that time had been upgraded to become a so called 'chief grammar school', namely one which offered eight years of teaching. This was precisely the school which later was attended by several famous mathematicians such as Johnny von Neumann and Eugene Wigner (or Jenó Pál Wigner as he was called at that time). Scholtz became the school director of the Lutheranian Grammar School in 1875. Unfortunately this excellent school was closed in 1952, and most of its equipment was lost. Due to the initiative and support of its former well-known students, among others Wigner, it was reopened in 1989 after being closed for thirty-seven years.

In 1879 Scholtz submitted his habilitation thesis

*Some theorems on the whole form of hexagrammum mysticum*(Hungarian) to the Faculty of Arts of the Hungarian Royal University of Budapest (the predecessor of the Loránd Eötvös University), where he became a docent. In 1878, his habilitation thesis was published by Atheneum Press. In 1884 he was promoted to full professor of mathematics, filling the chair Otto Petzval had occupied. At that time the university had four faculties: arts, law, theology and medicine. The mathematics department was, along with geology, in the faculty of arts.

At the Hungarian Royal University, Ágoston Scholtz was a colleague of Loránd Eötvös. They were often on the same committee set up to examine doctoral candidates, who usually offered mathematics, physics and astronomy as the subjects to be examined. Some famous Hungarian mathematicians took their doctoral examination in front of a committee on which Scholtz was one of the examiners of their main subject and a referee for their thesis. For example, three such mathematicians were József Kürschák in 1890, Frigyes Riesz in 1902, and Lipót Fejér also in 1902. Fejér also worked as an assistant in Scholtz' department from 1901. Several students who went on to become grammar school teachers wrote their doctoral theses with Ágoston Scholtz as their advisor.

Scholtz was one of the founders of the so called "table society of mathematicians" in 1885. The older initiators of the Society were Jeno Hunyadi and Kálmán Szily from the Royal Joseph Technical University and Ágoston Scholtz from the Royal Hungarian University, the middle generation was represented by Loránd Eötvös and Gyula König, the younger generation by Gusztáv Rados and Manó Beke. This society had no head and no basic documents or constitution. They had meetings twice a month aimed at mathematics and physics teachers from both universities and secondary schools. After listening to mathematics and physics lectures concerning the most interesting current developments, they would sit down to dinner (see [12]). Denes Konig writes (quoted in [4]):-

The more formal association, the Hungarian Mathematics and Physics Association, was founded in 1891. Ágoston Scholtz was a member of its main committee which consisted of twenty-four people.The Society had no presidents or statutes and their meetings often looked like dinner parties but for the solemn blackboard.

Scholtz's field of research was projective geometry and theory of determinants. His results were recorded by Muir in his famous work

*The history of determinants*and some of the most significant are presented in [4]. One of his main collaborators was Jeno Hunyadi who was six years older than Scholtz and died in 1889. They considered projective geometrical questions about conic sections and they transformed these questions into algebraic equations, where the determinant came into play. At that time it was usual to consider so called compound determinants where the entries of the matrix were also determinants themselves. In

*Six points lying on a conic section, and the theorem hexagrammum mysticum*(1877) and

*Sechs Punkte eines Kegelschnittes*(1878) he proved Pascal's theorem in Steiner's generality, by reducing it to an equation involving certain determinants. An intermediate step was to consider a matrix, which was later called the Hunyadi-Scholtz matrix. It is the matrix of a linear transformation induced on the vector space of homogeneous degree 2 polynomials by a linear substitution. In later years this was called the power transformation. In

*Six points lying on a conic section, and the theorem hexagrammum mysticum*(1877) and

*Sechs Punkte eines Kegelschnittes*Ⓣ (1878) it was proved for

*n*= 3 that the determinant of this transformed matrix is

*D*

^{n+1}, where

*D*is the determinant of the matrix of the linear substitution. See also Jeno Hunyadi's paper

*Beitrag zur Theorie der Flächen zweiten Grades*Ⓣ (1880).

He gave a lecture at the Hungarian Academy of Sciences on 5 February 1877 in which he explained these important results, see

*Six points lying on a conic section, and the theorem hexagrammum mysticum*(1877). The joint theorem of Hunyadi and Scholtz, the so called Hunyadi-Scholtz determinant theorem in the newer literature, was well known for several decades. Szenassy explains [4]:-

This theorem was used and generalized by several authors; see, for example, [9], [10], and [11].In spite of the ostensible complexity of generating the higher-order determinant, the Hunyadi-Scholtz theorem proved most useful in a variety of fields. Several mathematicians have applied it, for instance, in research into conic sections and second order surfaces in the cases n =2and n =3. Many scholars can be listed(e.g. Mertens, Pasch, Caspary, Müller, Szabó and others)who put it to good use in other questions of geometry. Its usefulness in technical problems is less well known: let us refer to Jeno Egervary's paper "Application of the Hunyadi-Scholtz matrices in the theory of grid structures"(1954)in which he studies grid structures with three bars constituting a triangle or six bars constituting a tetrahedron, the structure being loaded by a system of forces in equilibrium.

Scholtz published the paper

*Sechs Punkte eines Kegelschnittes*Ⓣ in the

*Archiv der Mathematik*in 1878, and around the same time several papers in Hungarian in the

*Müegyetemi Lapok*. For example

*Six points lying on a conic section, and the theorem hexagrammum mysticum*(1877),

*One theorem about determinants*(1877),

*Six points on a conic section and the theorem of Chasles*(1877), and

*Some determinant forms of covariant character*(1878). He also submitted solutions to problems that had been posed in this journal. Unfortunately this Hungarian language mathematics and physics journal only managed to survive for three volumes because of financial difficulties. Scholtz' later papers appeared in the

*Nouvelle Annales de Mathématique*, for example

*Résolution de l'equation du troisiéme degré*Ⓣ (1881), and in the

*Yearbook of the Grammar School in Igló*, see

*A remark on light interference*(1886).

In 1897 Scholtz took part in the first International Congress of Mathematicians held in Zürich, where two young Hungarian mathematicians were giving lectures (G Rados, and A Stodola). Scholtz retired from his chair in 1909. In 1911 Loránd Eötvös invited professor Lipót Fejér to become head of the department which had been led earlier by Scholtz.

Ágoston Scholtz died on 6 May 1916 in Veszprém. Further details of his life are given in [1], [2], [3], [4], [5], [6], [8], [7], [12], [13].

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

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