How was Neptune found with pencil and paper before anyone saw it?

Neptune was found by arithmetic before it was found by eye. In the 1840s astronomers noticed that Uranus was not following the path that Newton's law of gravity predicted. Two astronomers worked out where an unseen planet would have to be to tug Uranus off course, and on the night of 23–24 September 1846, an astronomer in Berlin pointed a telescope at the spot and found it, less than one degree from the prediction. François Arago said Urbain Le Verrier had discovered a planet with the point of his pen.

The puzzle began with Uranus itself. Discovered by William Herschel in 1781, it had nearly completed a full orbit by 1846, and irregularities in its path could not be explained by Newton's law alone. There were three suspects: gravity might behave differently far from the Sun, the observations might be wrong, or an undiscovered planet might be pulling on Uranus.

Working it backwards was brutal. It is an inverse problem: you know the effect and must find the cause. Today a computer does it in a blink, but in the 1840s it meant much laborious manual calculation. In 1845 John Couch Adams in Cambridge and Urbain Le Verrier in Paris separately began those calculations.

Le Verrier could not interest any French astronomer in looking, so he posted his results to Johann Galle at the Berlin Observatory. Galle found the planet that night. Galle replied to Le Verrier with astonishment: the planet whose place you have computed really exists.

The finding confirmed Newton's gravity far out at the edge of the Solar System. It also started a tense argument over who deserved the credit, which historians were still revisiting in 2003.