  Motivated by the integral representation of the Euler Beta function, we
introduce its Cauchy siblings and investigate some of their properties. Two of
these newly introduced functions happen to coincide with some classical means,
such as the arithmetic or the logarithmic Cauchy one. Although the bivariable
generalizations of Beta functions are obtained by elemantary integration, it
seems difficult to obtain closed formulas for more than two variables. % The
questions whether these Cauchy Beta functions belong to their respective class
of Cauchy quotients is addressed and answered positively in the case of the
Euler Beta function, but postponed to a future paper for all the other cases.
