Every calculation so far belonged to a world of two bodies. The solar system is not that.
The Earth is pulled not only by the Sun but by Venus and by Jupiter. Those planets come nearer and go farther, so their pull keeps changing in size and direction.
The pull of the other bodies is called perturbation: what has to be added to the two-body answer afterwards.
Include it and the problem is a three-body problem, with no closed solution. But because the extra pull is small, methods of successive approximation become available.
Take the size first. For the Earth, how does the pull of Jupiter compare with the pull of the Sun?
Jupiter is about a thousandth of the mass of the Sun, and even at closest approach it is more than four times as far away. The division gives roughly one part in twenty thousand.
So two-body answers usually hold. For a year of the motion of the Earth, ignoring Jupiter costs nothing that matters.
They stop holding after a long time. That is what makes perturbation troublesome.
The shift in a single lap is too small to see. But if it always shifts the same way, that changes things.
Shifts in scattered directions cancel out over time. Shifts that line up simply add, lap after lap.
A thousandth of a degree per lap is a whole degree after a thousand laps. A degree is visible to the naked eye, let alone a telescope.
Whether they line up is settled by the relation between the periods, and that is the subject of the next article.
After Uranus was found in , following its position showed that it did not go where the calculation said. The discrepancy was about arcminutes, easily visible to the observations of the day.
Le Verrier and Adams independently asked the same question: what planet, farther out, would be needed to displace Uranus by that much?
The calculation returned a mass, an orbit, and a direction in the sky. On September the Berlin observatory pointed a telescope that way and found Neptune the same night, within a degree of the prediction.
The position of a planet, named in advance by calculation alone. Newtonian gravity could hardly have been tested more severely.
Le Verrier turned the same method on Mercury, whose perihelion also carried arcseconds a century that the other planets could not account for.
The answer came out in the same form: there must be an unknown planet inside the orbit of Mercury. It was even given a name, Vulcan.
It was never found. Every eclipse was searched, shadows crossing the Sun were reported more than once, and none of it was ever confirmed.
Perturbation theory had not failed. What was insufficient was the law of gravity, and the answer came years later from general relativity. The same method found a planet in one case and pointed to a new theory of gravity in the other.