Which planet was found by mathematical prediction before anyone saw it?

Answer

Irregularities in Uranus's orbit led Urbain Le Verrier and John Couch Adams to predict a new planet. It was spotted in 1846 where Le Verrier calculated.

Context

After William Herschel discovered Uranus in 1781, astronomers found its motion did not match calculations based on the known planets. Alexis Bouvard's tables in 1821 could not be reconciled with later observations. Urbain Le Verrier in France, and independently John Couch Adams in England, proposed that gravity from an unseen planet was tugging Uranus, and each worked out where it should be.

Le Verrier sent his prediction to the Berlin Observatory, where Johann Galle and Heinrich d'Arrest searched on the night of September 23, 1846, and found a new object within about one degree of the predicted place. Adams had worked out a similar position a year earlier, but Cambridge and Greenwich were slow to act, and the Cambridge search recorded the planet without recognizing it. A bitter priority dispute followed, and credit is now usually shared.

The prediction was less exact than its fame suggests. The calculation assumed an orbit farther out than the real one, so the true distance, about 30 times the Earth-Sun distance, differed noticeably from the predicted one; the sky position happened to match because the planets were then near the same line. Galileo had recorded Neptune in 1612 and 1613 but took it for a fixed star. Neptune takes about 165 years to orbit and completed its first full circuit since discovery in 2011.