On translations in hyperbolic geometry of arbitrary (finite or infinite) dimension \(>1\)

Abstract. Hyperbolic geometry of the plane was discovered by J. Bolyai (1802–1860), C.F. Gauß (1777–1855), and N. Lobachevski (1793–1856). – In our book [3] we associate to every real vector space \(X\) of finite or infiinite dimension \(>1\), and equipped with a fixed inner product \(\delta:X\times X\to{\mathbb R}\), a hyperbolic geometry such that \((X,\delta),(X',\delta')\) are isomorphic if, and only if, the associated hyperbolic geometries are isomorphic. – In this paper we present a common treatment of translations in euclidean and hyperbolic geometry of arbitrary (finite or infinite) imension greater than one.