Gustav Ritter von Escherich (1 June 1849 – 28 January 1935) was an
Austria
Austria, , bar, Östareich officially the Republic of Austria, is a country in the southern part of Central Europe, lying in the Eastern Alps. It is a federation of nine states, one of which is the capital, Vienna, the most populous ...
n mathematician.
Biography
Born in
Mantua
Mantua ( ; it, Mantova ; Lombard and la, Mantua) is a city and '' comune'' in Lombardy, Italy, and capital of the province of the same name.
In 2016, Mantua was designated as the Italian Capital of Culture. In 2017, it was named as the Eur ...
, he studied mathematics and physics at the
University of Vienna
The University of Vienna (german: Universität Wien) is a public research university located in Vienna, Austria. It was founded by Duke Rudolph IV in 1365 and is the oldest university in the German-speaking world. With its long and rich hist ...
. From 1876 to 1879 he was professor at the
University of Graz
The University of Graz (german: link=no, Karl-Franzens-Universität Graz, ), located in Graz, Austria, is the largest and oldest university in Styria, as well as the second-largest and second-oldest university in Austria.
History
The univers ...
. In 1882 he went to the
Graz University of Technology
Graz University of Technology (german: link=no, Technische Universität Graz, short ''TU Graz'') is one of five universities in Styria, Austria. It was founded in 1811 by Archduke John of Austria and is the oldest science and technology research ...
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
Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of ther ...
and
Emil Müller he founded the
Austrian Mathematical Society
The Austrian Mathematical Society (german: Österreichische Mathematische Gesellschaft) is the national mathematical society of Austria and a member society of the European Mathematical Society.
History
The society was founded in 1903 by Ludwig B ...
.
Escherich died in
Vienna
en, Viennese
, iso_code = AT-9
, registration_plate = W
, postal_code_type = Postal code
, postal_code =
, timezone = CET
, utc_offset = +1
, timezone_DST ...
.
Work on hyperbolic geometry
Following
Eugenio Beltrami's (1868) discussion of
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
, Escherich in 1874 published a paper named "The geometry on surfaces of constant negative curvature". He used coordinates initially introduced by
Christoph Gudermann
Christoph Gudermann (25 March 1798 – 25 September 1852) was a German mathematician noted for introducing the Gudermannian function and the concept of uniform convergence, and for being the teacher of Karl Weierstrass, who was greatly influen ...
(1830) for spherical geometry, which were adapted by Escherich using
hyperbolic function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
s. For the case of translation of points on this surface of negative curvature, Escherich gave the following transformation on page 510:
:
and
which is identical with the relativistic
velocity addition formula by interpreting the coordinates as velocities and by using the
rapidity:
:
or with a
Lorentz boost
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
by using
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
:
:
These are in fact the relations between the coordinates of Gudermann/Escherich in terms of the
Beltrami–Klein model
In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk (or ''n'' ...
and the Weierstrass coordinates of the
hyperboloid model
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperbolo ...
- this relation was pointed out by
Homersham Cox (1882, p. 186),.
References
External links
*
{{DEFAULTSORT:Escherich, Gustav von
1849 births
1935 deaths
19th-century Austrian mathematicians
20th-century Austrian mathematicians
Austrian knights
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
Austro-Hungarian mathematicians