  In this note we approach the classical, Newtonian, gravitational $N$-body
problem by mean of a new, original numerical integration method. After a short
summary of the fundamental characteristics of the problem, including a sketch
of some of its mathematical and numerical issues, we present the new algorithm,
which is applied to a set of sample cases of initial conditions in the
`intermediate' $N$ regime ($N=100$). This choice of $N$ is not due to algorithm
limitation but just for computational convenience, in what this preliminary
work aims mainly to the presentation of the new method and so we wanted just to
provide an acceptable although statistical significant comparison with other
integration schemes. The proposed algorithm seems to be fast and precise at the
same time and so promising for further, more realistic, tests and scientific
applications.
