Gustav Ritter von Escherich
Born	1 June 1849 Mantua,Austrian Empire
Died	28 January 1935(1935-01-28) (aged 85) Vienna,Federal State of Austria
Citizenship	Austrian
Alma mater	University of Vienna (PhD, 1873)
Known for	Monatshefte für Mathematik und Physik Austrian Mathematical Society
Scientific career
Fields	Mathematics
Institutions	University of Vienna University of Graz Graz University of Technology
Thesis	Die Geometrie auf Flächen constanter negativer Krümmung (1873)
Doctoral advisor	Johannes Frischauf Karl Friesach
Doctoral students	Johann Radon

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

Born inMantua, he studied mathematics and physics at theUniversity of Vienna. From 1876 to 1879 he was professor at theUniversity of Graz. In 1882 he went to theGraz 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 withEmil Weyr he founded the journalMonatshefte für Mathematik und Physik and together withLudwig Boltzmann andEmil Müller he founded theAustrian Mathematical Society.

Escherich died inVienna.

FollowingEugenio Beltrami's (1868) discussion ofhyperbolic geometry, Escherich in 1874 published a paper named "The geometry on surfaces of constant negative curvature". He used coordinates initially introduced byChristoph Gudermann (1830) for spherical geometry, which were adapted by Escherich usinghyperbolic functions. For the case of translation of points on this surface of negative curvature, Escherich gave the following transformation on page 510:^[1]

${\displaystyle x=\frac{\sinh \frac{a}{k}+x^{\prime }\cosh \frac{a}{k}}{\cosh \frac{a}{k}+x^{\prime }\sinh \frac{a}{k}}}$ and${\displaystyle y=\frac{y^{\prime }}{\cosh \frac{a}{k}+x^{\prime }\sinh \frac{a}{k}}}$

which is identical with the relativisticvelocity addition formula by interpreting the coordinates as velocities and by using therapidity:

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

or with aLorentz boost by usinghomogeneous coordinates:

${\displaystyle (x,y,x^{\prime },y^{\prime })=\left(\frac{x_1}{x_0},\frac{x_2}{x_0},\frac{x_1^{\mathrm{\prime }}}{x_0^{\mathrm{\prime }}},\frac{x_2^{\mathrm{\prime }}}{x_0^{\mathrm{\prime }}}\right)}$

These are in fact the relations between the coordinates of Gudermann/Escherich in terms of theBeltrami–Klein model and the Weierstrass coordinates of thehyperboloid model - this relation was pointed out byHomersham Cox (1882, p. 186).^[2]

• Gustav von Escherich at theMathematics Genealogy Project

• 1849 births
• 1935 deaths
• Mathematicians from Austria-Hungary
• 19th-century Austrian mathematicians
• 20th-century Austrian mathematicians
• Scientists from Mantua
• University of Vienna alumni
• Academic staff of the University of Vienna
• Academic staff of the University of Graz
• Academic staff of Chernivtsi University

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