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.
Historical comet observations in China
A computer calculation of the past orbits of Halley using numerical integration could not continue past 837 AD because a very close approach to Earth made calculation of orbits prior to that too inaccurate. The researchers had started with accurate European measurements from 1759, 1682 and 1607 AD and calculated backwards. However, no accurate European records exist for 837 AD so the researchers looked to Chinese records. From these they obtained a good estimate of the perihelion of Halley for that year. They used that, together with perihelions found from Chinese data for 374 and 141 AD, to provide constraints for their calculation. They were finally able to calculate the orbits of Halley back to 1404 BC.