  Generally making the cosmological scale factor $R$ be a function of the
coordinate of the extra dimension $\sigma $ that is also a function of time
$t$, we achieve a new kind of cosmic acceleration mechanism depending on extra
dimension. We give the constraints on $\sigma $ under four different prospects
of the universe, and indicate that dark energy is not required for both the
small extra dimension and the accelerating expansion of our universe. This
results in this paper show that the accelerating expansion of our universe may
come from both dark energy and the achieved new mechanism here, or the latter
alone.
