Observational history of comets
Halley at first agreed with the longtime consensus that each comet was a different entity making a single visit to the solar system. In 1705, he applied Newton's method to 23 cometary apparitions that had occurred between 1337 and 1698. Halley noted that three of these, the comets of 1531, 1607, and 1682, had very similar orbital elements, and he was further able to account for the slight differences in their orbits in terms of gravitational perturbation by Jupiter and Saturn. Confident that these three apparitions had been three appearances of the same comet, he predicted that it would appear again in 1758–9. (Earlier, Robert Hooke had identified the comet of 1664 with that of 1618, while Giovanni Domenico Cassini had suspected the identity of the comets of 1577, 1665, and 1680. Both were incorrect.) Halley's predicted return date was later refined by a team of three French mathematicians: Alexis Clairaut, Joseph Lalande, and Nicole-Reine Lepaute, who predicted the date of the comet's 1759 perihelion to within one month's accuracy. Halley died before the comet's return; when it returned as predicted, it became known as Halley's Comet (with the latter-day designation of 1P/Halley). The comet next appears in 2061.
Comet Encke
As its official designation implies, Encke's Comet was the first periodic comet discovered after Halley's Comet (designated 1P/Halley). It was independently observed by several astronomers, the second being Caroline Herschel in 1795 and the third Jean-Louis Pons in 1818. Its orbit was calculated by Johann Franz Encke, who through laborious calculations was able to link observations of comets in 1786 (designated 2P/1786 B1), 1795 (2P/1795 V1), 1805 (2P/1805 U1) and 1818 (2P/1818 W1) to the same object. In 1819 he published his conclusions in the journal "Correspondance astronomique", and predicted correctly its return in 1822 (2P/1822 L1). It was recovered by Carl Ludwig Christian Rümker at Parramatta Observatory on 2 June 1822.