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 c ...

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 Europ ...

, 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 histo ...

. 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 thermodyn ...

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 ...

s. For the case of translation of points on this surface of negative curvature, Escherich gave the following transformation on page 510: :

x=\frac

and

y=\frac

which is identical with the relativistic

velocity addition formula In relativistic physics, a velocity-addition formula is a three-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive Lorentz transformations, so they also relate different fr ...

by interpreting the coordinates as velocities and by using the

rapidity In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with dis ...

: :

\frac=\tanh\frac=\frac

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 ...

: :

(x,\ y,\ x',\ y')=\left(\frac,\ \frac,\ \frac,\ \frac\right)

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 hyperboloid ...

- 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