The first planets discovered since ancient times were Uranus and Neptune How were they discovered?

Answer 1

Jupiter and Saturn are visible to the naked eye and have thus been know since ancient times (thus nobody knows who first discovered them).
Although Uranus is visible to the naked eye, it was never recognized as a planet by ancient observers because of its dimness and slow orbit. Sir William Herschel announced its discovery on March 13, 1781.
Neptune was the first planet found by mathematical prediction rather than by observation. Unexpected changes in the orbit of Uranus led Alexis Bouvard to deduce that its orbit was subject to gravitational perturbation by an unknown planet. Neptune was subsequently observed on September 23, 1846 by Johann Galle within a degree of the position predicted by Urbain Le Verrier.

Answer 2

Uranus was discovered by Sir William Herschel on March 13th 1781, while looking at the sky with his telescope, at first he though it was a comet, but later observations showed it was a planet.

With precise measurements, it was discovered that Uranus had weak disturbances in its orbit, and so it was mathematically predicted that another planet is beyond it - and so Neptune was discovered where the calculations said so.