  We present an analytical proof assisted by computer calculations for the
dynamical stability of the eight main planets and Pluto for the next 100,000
years. It means that the semi-major axes of the planets will not change
significantly during this period. Also the eccentricities and inclinations of
the orbits will remain sufficiently small. A standard linear four-step
numerical method is used to integrate approximately the orbits of Mercury,
Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto. Written in
orbital elements, the dynamics of the nine planets manifests a system of 54
first-order ordinary differential equations. The step-size of the numerical
method -- about six days, has been performed 6,290,000 times. We estimate the
total accumulation of rounding-off errors, deviations related to possible
uncertainty in the astronomical data and the accuracy of the computer
calculations.
