When was Neptune discovered?
Short answer: Over two nights on September 23rd & 24th, 1846.
Long answer: Neptune is the first planet to be discovered using mathematics. Here's what happened...
Since discovering the planet Uranus in 1781, astronomers had been watching its orbit and they noticed that it seemed to speed up and slow down slightly as it travelled around the Sun. The only way to explain this would be another planet farther out that was pulling on Uranus as it passed by.
French mathematician Urbain Le Verrier figured out where the new planet should be, and he showed his calculations to astronomer Johann Gottfried Galle. When Galle started looking in the right part of the sky, he found Neptune almost immediately.
This was a very big deal. It was amazing proof that people could understand and even predict the sizes and movements of celestial bodies.
More information: Who discovered Neptune?