  In this chapter, dedicated to the 60th Anniversary of Themistocles M.
Rassias, M\"obius transformation and Einstein velocity addition meet in the
hyperbolic geometry of Bolyai and Lobachevsky. It turns out that M\"obius
addition that is extracted from M\"obius transformation of the complex disc and
Einstein addition from his special theory of relativity are isomorphic in the
sense of gyrovector spaces.
