!!!Mathematik

Mathematics: A translator of mathematical and astronomical texts, 
Hermann von Kaernten became the first outstanding Austrian 
mathematician during the first half of the 12%%sup th/%  century. The 
science of mathematics began to flourish in Austria in the 
15%%sup th/%  century, when the first School of Mathematicians was 
founded at Vienna University. Its main representatives were Johannes 
von Gmunden, Georg von Peuerbach and Regiomontanus (J. Mueller). They 
made mathematics an ancillary science of astronomy (methods of 
calculation, trigonometry, revision of astronomical tables). During 
the 16%%sup th/%  century the second Vienna School of Mathematicians 
began to form, the main representatives of which were J. Stabius, S. 
Rosinus, G. Tannstetter, J. Voegelin, H. Schreiber and C. Rudolff. 
These remarkable men made major contributions to arithmetic and 
algebra (most of them were members of an association of mathematicians 
teaching basic mathematics to traders and merchants in the 
15%%sup th/%  and 16%%sup th/%  centuries), to cartography, geometry 
and the calculation of cubic capacity. The first university professors 
of mathematics were appointed by Emperor Maximilian I at Vienna 
University. G. J. v. Lauchen (also called Rheticus), born at 
Feldkirch, Vorarlberg, wrote the mathematical introduction to the work 
of Copernicus. J Kepler lectured at the Universities of Graz (from 
1594), Prague and Linz (from 1612). He used conic sections for 
astronomical calculations (Kepler´s laws), calculated tables 
concerning the motion of planets (Tabulae Rudolphinae) and worked out 
tables of logarithms (in cooperation with Jost Buergi; proved their 
correctness by means of algebraic calculations). Together with W. 
Schickhart, he invented the first mechanical calculating machine. His 
approach to the problem of calculating cubic capacity ("Nova 
Stereometria doliorum vinariorum") made him one of the pioneers in the 
field of infinitesimal calculus. The eminent mathematicians of the 
16%%sup th/%  and 17%%sup th/%  centuries were to be found among 
Jesuit monks, e.g. P. Gudin, who intensely occupied himself with the 
calculation of cubic capacity and lectured at the Universities of 
Vienna and Graz; C. Grienberger, born at Hall in Tirol, who lectured 
at Graz before he was offered a chair at Rome University. Other 
outstanding mathematicians among the Jesuits worked as missionaries 
(e.g. J. Grueber).

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In the 19%%sup th/%  century J. Petzval lectured at Vienna University. 
He managed to work out the Laplace transform with which differential 
equations could be solved. Because of a conflict with S. Spitzer, 
however, the transform was named after the French mathematician 
Laplace. Petzval developed high-speed lenses, cooperated with the 
famous Austrian optician, Voigtlaender, and invented a special lens 
for portrait photography. He also made important contributions to 
mathematical optics, ballistics and music theory. A general reform of 
the universities enforced in 1852 led to mathematics reaching another 
zenith in the second half of the 19%%sup th/%  century. As well as 
Petzval, E. Weyr and G. v. Escherich, the founders of the leading 
publication on mathematics in Austria (Monatshefte fuer Mathematik und 
Physik), worked at Vienna University. Other remarkable mathematicians 
of that time were L. Koenigsberger, F. Mertens and L. Gegenbauer 
(introduced a course on actuarial theory at Vienna University).

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A. v. Burg and J. Salomon lectured at the Vienna University of 
Technology, which had been founded in 1815. The latter wrote textbooks 
which became extremely popular with students of mathematics. He also 
worked out the algorithm of subtraction, which since that time has 
been referred to as "Austrian method". K. Zsigmondy studied number 
theory. W. Czuber concerned himself with statistics, probability 
calculus and actuarial theory. O. Stolz (responsible for the 
dissemination of modern differential and integral calculus) and K. 
Zindler (expert in geometry; later professor at Graz) taught at 
Innsbruck University. R. Sterneck (number theory, tidology) and J. 
Frischauf (spherical functions, geodesy and cartography) lectured at 
Graz University.

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During the 20%%sup th/%  century numerous outstanding Austrian 
mathematicians appeared on the scene. Some of them held a chair at 
Vienna University, e.g. W. Wirtinger (function theory, third 
mathematician to be awarded the Cayley Award after Cantor and 
Poincare), H. Hahn (one of the originators of functional analysis, 
Hahn-Banach´s theorem), P. Furtwaengler (one of the 
co-originators of the Vienna School of Number Theory), J. Radon 
(theory of measure and content, integral theory), N. Hofreiter 
(geometry of numbers, algebra), E. Hlawka (number theory, managed to 
verify a hypothesis by Minkowski), K. Goedel (basic mathematics) and 
K. Menger (topology, dimension theory). Another well-known 
mathematician, A. Tauber (Tauber´s theorems) belonged to the 
teaching staff of both the Vienna University of Technology and Vienna 
University. Other notable professors of Vienna University were H. 
Tietze, E. Helly, L. Vietoris (one of the originators of general 
topology) and O. Schreier (algebra). It should also be mentioned that 
E. Mueller, J. Krames and W. Wunderlich founded a centre of 
descriptive geometry at the Vienna University of Technology.

!Literature
H. Kaiser and W. Noebauer, Geschichte der Mathematik, 
1984; W. Noebauer, Geschichte der Mathematik in Oesterreich in: 
Wissenschaftliche Nachrichten 46, 1978; H. Kaiser, Die Rolle 
Oesterreichs in der Geschichte der Mathematik, in: Didaktikhefte der 
Oesterreichischen mathematischen Gesellschaft, 1989.


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