  The study of the differences detected between the observed and the predicted
positions of Uranus taking only the ancient planets into account led to the
discovery of planet Neptune in 1846. This event remains one of the best
accomplishments ever achieved in the history of Astronomy and Classical
Mechanics. In this paper, we study the perturbations in the orbit of Uranus due
to Neptune and its effects from a modern numerical point of view of the
$N$-body problem. The effects induced by Pluto in the orbit of Neptune, as the
historical search for a ninth planet in the Solar System (recently boostered
again with the hypothesis of the so-called Planet Nine) back in the days was
propelled by some supposed small inconsistencies in the orbit of the ice
giants, are also analyzed.
