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.
Eilmer of Malmesbury
If Eilmer had seen Halley's comet seventy-six years earlier in 989, he could have been born about 984, making him about five or six years old when he first saw the comet, old enough to remember it. However the periodicity of comets was probably unknown in Eilmer's time, and so his remark "It is long since I saw you" could have referred to a different, later comet. Since it is known that Eilmer was an "old man" in 1066, and that he had made the flight attempt "in his youth", the event is placed some time during the early 11th century, perhaps in its first decade.