Gustav von Escherich

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.

{{further|History of Lorentz transformations}}

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:{{Cite journal|author=Escherich, G. von|year=1874|title=Die Geometrie auf den Flächen constanter negativer Krümmung|journal=Wiener Sitzungsberichte IIA|volume=69|pages=497–526|url=https://books.google.com/books?id=6e0DAAAAIAAJ&pg=PA497}}

:$x=\backslash frac\{\backslash sinh\backslash frac\{a\}\{k\}+x\text{'}\backslash cosh\backslash frac\{a\}\{k\}\}\{\backslash cosh\backslash frac\{a\}\{k\}+x\text{'}\backslash sinh\backslash frac\{a\}\{k\}\}$ and $y=\backslash frac\{y\text{'}\}\{\backslash cosh\backslash frac\{a\}\{k\}+x\text{'}\backslash sinh\backslash frac\{a\}\{k\}\}$

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

:$\backslash frac\{\backslash sinh\backslash frac\{a\}\{k\}\}\{\backslash cosh\backslash frac\{a\}\{k\}\; \}=\backslash tanh\backslash frac\{a\}\{k\}=\backslash frac\{v\}\{c\}$

or with a Lorentz boost by using homogeneous coordinates:

:$(x,\backslash \; y,\backslash \; x\text{'},\backslash \; y\text{'})=\backslash left(\backslash frac\{x\_\{1\}\}\{x\_\{0\}\},\backslash \; \backslash frac\{x\_\{2\}\}\{x\_\{0\}\},\backslash \; \backslash frac\{x\_\{1\}^\{\backslash prime\}\}\{x\_\{0\}^\{\backslash prime\}\},\backslash \; \backslash frac\{x\_\{2\}^\{\backslash prime\}\}\{x\_\{0\}^\{\backslash prime\}\}\backslash right)$

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).{{Cite journal|author=Cox, H.|year=1881|title=Homogeneous coordinates in imaginary geometry and their application to systems of forces|journal=The Quarterly Journal of Pure and Applied Mathematics|volume=18|issue=70|pages=178–192|url=http://resolver.sub.uni-goettingen.de/purl?PPN600494829_0018}}

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• {{MathGenealogy|id=27259}}

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Category:1849 births

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Category:Mathematicians from Austria-Hungary

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