How was Neptune found?

By arithmetic. Uranus was not keeping to its predicted orbit, and Urbain Le Verrier calculated where an unseen planet would have to sit to pull it off course. He sent the coordinates to the Berlin Observatory, and Johann Galle found it the same night, within a degree of the prediction - which remains one of the most spectacular confirmations any theory has received.

John Couch Adams in Cambridge had done substantially the same calculation earlier and been largely ignored by British astronomers, which produced a long and unedifying priority row between the two countries. The modern view gives them both credit and reserves some criticism for the Astronomer Royal, who had Adams's figures and did not act on them.

There is a footnote that spoils the romance slightly. Galileo drew Neptune twice in 1612 and 1613 while observing Jupiter's moons, recorded it as a fixed star, and on one night noted that it appeared to have moved - and did not follow it up. He had the observation, the instrument and the curiosity, and missed a planet by a fortnight of attention.