Editing Hyperbolic coordinates (section)

==History==
The [[geometric mean]] is an ancient concept, but [[hyperbolic angle]] was developed in this configuration by [[Gregoire de Saint-Vincent]]. He was attempting to perform [[quadrature (mathematics)|quadrature]] with respect to the rectangular hyperbola ''y'' = 1/''x''. That challenge was a standing [[open problem]] since [[Archimedes]] performed the [[quadrature of the parabola]]. The curve passes through (1,1) where it is opposite the [[origin (mathematics)|origin]] in a [[unit square]]. The other points on the curve can be viewed as [[rectangle]]s having the same [[area]] as this square. Such a rectangle may be obtained by applying a [[squeeze mapping]] to the square. Another way to view these mappings is via [[hyperbolic sector]]s. Starting from (1,1) the hyperbolic sector of unit area ends at (e, 1/e), where [[e (mathematical constant)|e]] is 2.71828…, according to the development of [[Leonhard Euler]] in ''[[Introduction to the Analysis of the Infinite]]'' (1748).

Taking (e, 1/e) as the vertex of rectangle of unit area, and applying again the squeeze that made it from the unit square, yields <math>(e^2, \ e^{-2}).</math> Generally n squeezes yields <math>(e^n, \ e^{-n}).</math> [[A. A. de Sarasa]] noted a similar observation of G. de Saint Vincent, that as the abscissas increased in a [[geometric series]], the sum of the areas against the hyperbola increased in [[arithmetic series]], and this property corresponded to the '''logarithm''' already in use to reduce multiplications to additions. Euler’s work made the [[natural logarithm]] a standard mathematical tool, and elevated mathematics to the realm of [[transcendental function]]s. The hyperbolic coordinates are formed on the original picture of G. de Saint-Vincent, which provided the quadrature of the hyperbola, and transcended the limits of [[algebraic function]]s.

In 1875 [[Johann von Thünen]] published a theory of natural wages<ref>{{cite book|author=Henry Ludwell Moore|author-link=Henry Ludwell Moore|title=Von Thünen's Theory of Natural Wages|url=https://archive.org/details/vonthnenstheor00moor|year=1895|publisher=G. H. Ellis}}</ref> which used geometric mean of a subsistence wage and market value of the labor using the employer's capital.

In [[special relativity]] the focus is on the 3-dimensional [[hypersurface]] in the future of spacetime where various velocities arrive after a given [[proper time]]. Scott Walter<ref>Walter (1999) page 99</ref> explains that in November 1907 [[Hermann Minkowski]] alluded to a well-known three-dimensional hyperbolic geometry while speaking to the Göttingen Mathematical Society, but not to a four-dimensional one.<ref>Walter (1999) page 100</ref>
In tribute to [[Wolfgang Rindler]], the author of a standard introductory university-level textbook on relativity, hyperbolic coordinates of spacetime are called [[Rindler coordinates]].