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.
Sungrazing comet
One of the first comets to have its orbit computed was the sungrazing comet (and Great Comet) of 1680, now designated C/1680 V1. It was observed by Isaac Newton and he published the orbit results in 1687. Later, in 1699, Jacques Cassini proposed that comets could have relatively short orbital periods and that C/1680 V1 was the same as a comet observed by Tycho Brahe in 1577, but in 1705 Edmond Halley determined that the difference between the perihelion distances of the two comets was too great for them to be the same object. However, this marked the first time that it was hypothesized that Great Comets were related or perhaps the same comet. Later, Johann Franz Encke computed the orbit of C/1680 V1 and found a period of approximately 9000 years, leading him to conclude that Cassini's theory of short period sungrazers was flawed. C/1680 V1 had the smallest measured perihelion distance until the observation in 1826 of comet C/1826 U1.