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

Austria Austria, formally the Republic of Austria, is a landlocked country in Central Europe, lying in the Eastern Alps. It is a federation of nine Federal states of Austria, states, of which the capital Vienna is the List of largest cities in Aust ...

n mathematician.

Biography

Born in

Mantua Mantua ( ; ; Lombard language, Lombard and ) is a ''comune'' (municipality) in the Italian region of Lombardy, and capital of the Province of Mantua, eponymous province. In 2016, Mantua was designated as the "Italian Capital of Culture". In 2 ...

, he studied mathematics and physics at the

University of Vienna The University of Vienna (, ) is a public university, public research university in Vienna, Austria. Founded by Rudolf IV, Duke of Austria, Duke Rudolph IV in 1365, it is the oldest university in the German-speaking world and among the largest ...

. From 1876 to 1879 he was professor at the

University of Graz The University of Graz (, formerly: ''Karl-Franzens-Universität Graz'') is a public university, public research university located in Graz, Austria. It is the largest and oldest university in Styria, as well as the second-largest and second-old ...

. 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 Ludwig Eduard Boltzmann ( ; ; 20 February 1844 – 5 September 1906) was an Austrian mathematician and Theoretical physics, theoretical physicist. His greatest achievements were the development of statistical mechanics and the statistical ex ...

and Emil Müller he founded the

Austrian Mathematical Society The Austrian Mathematical Society () 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 Boltzmann, Gustav von Escherich, and Emil Müller ...

. Escherich died in

Vienna Vienna ( ; ; ) is the capital city, capital, List of largest cities in Austria, most populous city, and one of Federal states of Austria, nine federal states of Austria. It is Austria's primate city, with just over two million inhabitants. ...

Work on hyperbolic geometry

Following

Eugenio Beltrami Eugenio Beltrami (16 November 1835 – 18 February 1900) was an Italian mathematician notable for his work concerning differential geometry and mathematical physics. His work was noted especially for clarity of exposition. He was the first to ...

's (1868) discussion of

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...

, 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 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 by interpreting the coordinates as velocities and by using the

rapidity In special relativity, the classical concept of velocity is converted to rapidity to accommodate the limit determined by the speed of light. Velocities must be combined by Einstein's velocity-addition formula. For low speeds, rapidity and velo ...

: :

\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 ...

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 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 hyperboloi ...

- this relation was pointed out by Homersham Cox (1882, p. 186).

References

External links

* {{DEFAULTSORT:Escherich, Gustav von 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