The funny part is that the "stability" of the celestial system is what convinced Western Europe of a great many scientific facts, and was one of the driving forces of the enlightenment. Even recently, one of the tests of relativity theory (determining whether light really does have a finite speed to be exact) was done by looking at the interference patterns generated by Jupiter's moons. The funny thing about that is that if you were to redo that experiment today you would get totally different values. The same is true for earlier theories, like Kepler's laws or Newton's gravity. The measurements that they purported to explain, they could not actually explain. But very short subsets of those measurements followed the discovered laws to near-perfection.
So the solar system, that convinced the Western world that only a "mechanical God" or no God at all ruled the heavens, because everything is perfectly predictable ... turns out not to be predictable, fickle like the weather. They didn't know that at the time, because reliable measurements had just started mere decades earlier, and in the short term they are extremely stable. But the random perturbations of the system (ie. comets just happening to pass by, modifying moon and planetary orbits while they swing by) have an influence that is so big that they destroy whatever signal is in the measurements. Planetary orbits are not stable at all, nor are they even remotely elliptical in practice (they are perturbed too much).
And it's worse yet than merely those comets, the fact is that it was not yet known that the "three body problem" is chaotic, and the relation between starter positions and "end" states is completely unpredictable. So a 11 huge bodies, trillions of smaller bodies system, like the solar system, is completely unstable on all but the shortest time frames. Now, even though we have better scientific models and more accurate theories, we have also found that actually predicting planetary orbits or moon orbits as little as 100 years out is completely impossible to do with any accuracy. This is because of "black swan" events with huge influence, that occur with alarming regularity.
In a way, it's funny, that we were convinced of pretty much the central claim of the enlightenment ... by a measurement error. By a failure to detect omnipresent chaos. The people that convinced the world that physical laws, not God, ruled the heavens ... were wrong. Not about the principle, those laws do apply, but about the actual examples they gave. The laws about gravity do not really predict planetary or moon orbits with any accuracy if there's constantly trillions of disturbances happening everywhere that you cannot measure and thus cannot implement in the calculations. It was just really hard to prove them wrong in the short term. And people believed them, and of course they did not modify their beliefs once it did become clear they were wrong. In this case it was a positive force, of course, but that was mere luck.
The sad part is that this lesson hasn't sunk in. Predicting past events to near-perfect accuracy is easy in a chaotic system. Predicting the future, or even just measuring the present situation, is not impossible, but so absurdly hard that we can safely say that no technology or theory will ever be able to do it.
I'm kind of afraid that's the next lesson we'll have to learn the hard way. Just because we have correct theories does not mean we can accurately predict the future. Chaos is everywhere. Not just in the heavens, but on earth as well, from wall street to the climate.