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Gustav von Escherich

Gustav von Escherich Austrian mathematician

Austrian mathematician
The basics
Quick Facts
Intro Austrian mathematician
Known for Monatshefte fuer Mathematik
A.K.A. Эшерих, Густав фон
Was Mathematician Professor Educator
From Austria
Type Academia Mathematics
Gender male
Birth 1 June 1849, Mantua, Kingdom of Italy
Death 28 January 1935, Vienna, Austria (aged 85 years)
Star sign GeminiGemini
Residence Austria, Austria
The details
Biography

Gustav Ritter von Escherich (1 June 1849 – 28 January 1935) was an Austrian mathematician.

Biography

Born in Mantua, he studied mathematics and physics at the University of Vienna. From 1876 to 1879 he was professor at the University of Graz. In 1882 he went to the Graz University of Technology and in 1884 he went to the University of Vienna, where he also was president of the university in 1903/04.

Together with Emil Weyr he founded the journal Monatshefte für Mathematik und Physik and together with Ludwig Boltzmann and Emil Müller he founded the Austrian Mathematical Society.

Escherich died in Vienna.

Work on hyperbolic geometry

Following Eugenio Beltrami's (1868) discussion of hyperbolic geometry, Escherich in 1874 published a paper named "The geometry on surfaces of constant negative curvature". He used coordinates initially introduced by Christoph Gudermann (1830) for spherical geometry, which were adapted by Escherich using hyperbolic functions. For the case of translation of points on this surface of negative curvature, Escherich gave the following transformation on page 510:

x = sinh a k + x cosh a k cosh a k + x sinh a k {\displaystyle x={\frac {\sinh {\frac {a}{k}}+x'\cosh {\frac {a}{k}}}{\cosh {\frac {a}{k}}+x'\sinh {\frac {a}{k}}}}} and y = y cosh a k + x sinh a k {\displaystyle y={\frac {y'}{\cosh {\frac {a}{k}}+x'\sinh {\frac {a}{k}}}}}

which is identical with the relativistic velocity addition formula by interpreting the coordinates as velocities and by using the rapidity:

sinh a k cosh a k = tanh a k = v c {\displaystyle {\frac {\sinh {\frac {a}{k}}}{\cosh {\frac {a}{k}}}}=\tanh {\frac {a}{k}}={\frac {v}{c}}}

or with a Lorentz boost by using homogeneous coordinates:

( x ,   y ,   x ,   y ) = ( x 1 x 0 ,   x 2 x 0 ,   x 1 x 0 ,   x 2 x 0 ) {\displaystyle (x,\ y,\ x',\ y')=\left({\frac {x_{1}}{x_{0}}},\ {\frac {x_{2}}{x_{0}}},\ {\frac {x_{1}^{\prime }}{x_{0}^{\prime }}},\ {\frac {x_{2}^{\prime }}{x_{0}^{\prime }}}\right)}

These are in fact the relations between the coordinates of Gudermann/Escherich in terms of the Beltrami–Klein model and the Weierstrass coordinates of the hyperboloid model - this relation was pointed out by Homersham Cox (1882, p. 186), see History of Lorentz transformations#Escherich.

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Reference sources
References
https://books.google.com/books?id=6e0DAAAAIAAJ&pg=PA497
http://resolver.sub.uni-goettingen.de/purl?PPN600494829_0018
https://www.genealogy.math.ndsu.nodak.edu/id.php?id=27259
https://d-nb.info/gnd/116569352
https://genealogy.math.ndsu.nodak.edu/id.php?id=27259
https://viaf.org/viaf/3225743
https://www.worldcat.org/identities/viaf-3225743
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