Following
Eugenio Beltrami's (1868) discussion of
hyperbolic geometry, 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 functions. For the case of translation of points on this surface of negative curvature, Escherich gave the following transformation on page 510: :x=\frac{\sinh\frac{a}{k}+x'\cosh\frac{a}{k}}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}} and y=\frac{y'}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}} which is identical with the relativistic
velocity addition formula by interpreting the coordinates as velocities and by using the
rapidity: :\frac{\sinh\frac{a}{k}}{\cosh\frac{a}{k} }=\tanh\frac{a}{k}=\frac{v}{c} or with a
Lorentz boost by using
homogeneous coordinates: :(x,\ y,\ x',\ y')=\left(\frac{x_{1}}{x_{0}},\ \frac{x_{2}}{x_{0}},\ \frac{x_{1}^{\prime}}{x_{0}^{\prime}},\ \frac{x_{2}^{\prime}}{x_{0}^{\prime}}\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 - this relation was pointed out by
Homersham Cox (1882, p. 186). ==References==