  A new modulus of smoothness based on the Euler angles is introduced on the
unit sphere and is shown to satisfy all the usual characteristic properties of
moduli of smoothness, including direct and inverse theorem for the best
approximation by polynomials and its equivalence to a $K$-functional, defined
via partial derivatives in Euler angles. The set of results on the moduli on
the sphere serves as a basis for defining new moduli of smoothness and their
corresponding $K$-functionals on the unit ball, which are used to characterize
the best approximation by polynomials on the ball.
