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.
Johann Franz Encke
Following a suggestion by Jean-Louis Pons, who suspected one of the three comets discovered in 1818 to be the same one already discovered by him in 1805, Encke began to calculate the orbital elements of this comet. At this time, all the known comets had an orbital period of seventy years and more, with an aphelion far beyond the orbit of Uranus. The most famous comet of this family was Comet Halley with its period of seventy-six years. So the orbit of the comet discovered by Pons was a sensation, because his orbit was found to have a period of 3.3 years, so that the aphelion had to be within the orbit of Jupiter. Encke predicted its return for 1822, but this return was observable only from the southern hemisphere and was seen by Carl Ludwig Christian Rümker in Australia. The comet was also identified with the one seen by Pierre Méchain in 1786 and by Caroline Herschel in 1795.