exactly the same phenomenon of successively sampling less and less significant digits still occurs. And once again, at least for a while, any randomness in the motion of the ball can be attributed to randomness in this initial digit sequence.
But after at most ten or so collisions, many other effects, mostly associated with continual interaction with the environment, will always in practice become important, so that any subsequent randomness cannot solely be attributed to initial conditions.
And indeed in any system, the amount of time over which the details of initial conditions can ever be considered the dominant source of randomness will inevitably be limited by the level of separation that exists between the large-scale features that one observes and small-scale features that one cannot readily control.
So in what kinds of systems do the largest such separations occur? The answer tends to be systems in astronomy. And as it turns out, the so-called three-body problem in astronomy was the very first place where sensitive dependence on initial conditions was extensively studied.
The three-body problem consists in determining the motion of three bodies—such as the Earth, Sun and Moon—that interact through gravitational attraction. With just two bodies, it has been known for nearly four hundred years that the orbits that occur are simple ellipses or hyperbolas. But with three bodies, the motion can be much more complicated, and—as was shown at the end of the 1800s—can be sensitively dependent on the initial conditions that are given.
The pictures on the next page show a particular case of the three-body problem, in which there are two large masses in a simple elliptical orbit, together with an infinitesimally small mass moving up and down through the plane of this orbit. And what the pictures demonstrate is that even if the initial position of this mass is changed by just one part in a hundred million, then within 50 revolutions of the large masses the trajectory of the small mass will end up being almost completely different.
So what happens in practice with planets and other bodies in our solar system? Observations suggest that at least on human timescales most of their motion is quite regular. And in fact this regularity was in