Lobachevsky geometry and stellar parallaxes

Using Beltrami–Poincaré models in the Euclidean semiplane and semispace for two-dimensional and three-dimensional Lobachevsky spaces, we give a simple deduction of some equations and inequalities by Lobachevsky from his first published work “On the foundations of geometry.” Lobachevsky had applied the equations and inequalities together with the stellar parallaxes then available to him to the question whether his “theory of parallels holds or fails in nature.” We also list some applications of Lobachevsky geometry.