The discovery of the planet Neptune was made partly by mathematical calculation. Scientists noticed that the motion of Uranus failed to exactly follow its predicted path across the sky. An English astronomer John Couch Adams and a French mathematician Urbain Le Verrier came to the conclusion that an undiscovered planet’s gravity must be affecting the motion of the planet Uranus.
Le Verrier calculated where this unknown planet might be in the sky. Using those predictions, Johann Galle of the Berlin Observatory found Neptune in 1846 in the predicted area on the first night that he looked for it.
Galileo also observed Neptune in 1612 and 1613 with his small telescope. He recorded it but thought it was a fixed star so that Neptune went undiscovered for another 233 years.
Neptune is the eighth planet from the sun and is one of four gas giants in the solar system composed largely of hydrogen and helium. Neptune is far from the sun making it too faint to see without optical aid.
During October it is located in the southeastern sky in the constellation of Aquarius the Water Bearer. If you own a small telescope you may be able to observe it as a faint, bluish, star-like object moving slowly against the background of the stars.