Unlike other planets that were found by empirical observation, Neptune was found by mathematical prediction involving Newton’s law of universal gravitation.
In 1821, Alexis Bouvard calculated future Uranus’ orbital position using Newton’s law of motion and gravitation. But according to actual observation, the orbit was slightly different from the expected position, leading Bouvard to predict the existence of an unknown celestial body perturbing Uranus’ orbit.
In 1845-1846, Urbain Le Verrier and John Couch Adams independently calculated the position, mass and orbit of the perturbing body.
On September 24, 1846, Johann Gottfried Galle searched the sky to find this hypothesized planet. Neptune was found only in an hour of search and within 1° from Le Verrier’s prediction.
In Francois Arago’s word, Le Verrier had discovered a planet with the point of his pen. The discovery of Neptune dramatically confirmed the validity of the law of universal gravitation.