The existence of the planet Neptune was actually predicted mathematically before it was directly observed. In the early 19th century, astronomers noticed irregularities in the orbit of Uranus that could not be explained solely by the gravitational influence of known celestial bodies. This led to the hypothesis that there must be another planet exerting gravitational forces on Uranus.
Using mathematical calculations, astronomers Urbain Le Verrier and John Couch Adams independently predicted the position of this hypothetical planet. In 1846, German astronomer Johann Galle observed the region of the sky indicated by Le Verrier's calculations and discovered Neptune in that exact position. This successful observation confirmed the existence of the predicted planet.
Since then, further observations and studies have provided additional evidence for the existence of Neptune. Astronomers have used telescopes and other instruments to study the planet's physical characteristics, such as its size, composition, and atmosphere. Space missions like NASA's Voyager 2 and the Hubble Space Telescope have provided detailed images and data about Neptune.
Additionally, the gravitational influence of Neptune has been observed on other celestial bodies, such as its moons and other planets. These observations further support the existence of Neptune and its gravitational interactions within the solar system.
In summary, the existence of Neptune was initially inferred through mathematical calculations that explained the orbital irregularities of Uranus. Subsequent observations, space missions, and gravitational effects on other objects have provided ample evidence to confirm the presence of the planet Neptune in our solar system.