Neptune was first spotted on this night in 1846. This was the first time that Newton's theory of gravitation had been used to deduce the position of an unknown planet.
In the early 19th century, astronomers noticed that Uranus did not move around the sun exactly as predicted. This suggested that an unknown object was orbiting the sun beyond Uranus, tugging on it. In 1845, the British astronomer John Couch Adams computed the position of the mystery object and asked James Challis, director of Cambridge Observatory, to check for a planet. Challis was dubious, as he later wrote: "It was so novel a thing to undertake observations in reliance upon merely theoretical deductions; and that while much labour was certain, success appeared very doubtful." He looked briefly, but missed the planet.
Meanwhile, French astronomer Urbain Jean Joseph Le Verrier was making similar calculations. In 1846 he sent his predictions to Johann Gottfried Galle, an astronomer at the Observatory of Berlin. It took less than half an hour for Galle to find an object not on his star chart. Observing its motion the following night, he confirmed it was a planet.
For a detailed history of Neptune's discovery, see www-history.mcs.st-andrews.ac.uk/history/HistTopics/Neptune_and_Pluto.html