  In this work a series of methods are developed for understanding the
Friedmann equation when it is beyond the reach of the Chebyshev theorem. First
it will be demonstrated that every solution of the Friedmann equation admits a
representation as a roulette such that information on the latter may be used to
obtain that for the former. Next the Friedmann equation is integrated for a
quadratic equation of state and for the Randall--Sundrum II universe, leading
to a harvest of a rich collection of new interesting phenomena. Finally an
analytic method is used to isolate the asymptotic behavior of the solutions of
the Friedmann equation, when the equation of state is of an extended form which
renders the integration impossible, and to establish a universal exponential
growth law.
