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.
Crab Nebula
The Crab Nebula was first identified in 1731 by John Bevis. The nebula was independently rediscovered in 1758 by Charles Messier as he was observing a bright comet. Messier catalogued it as the first entry in his catalogue of comet-like objects; in 1757, Alexis Clairaut reexamined the calculations of Edmund Halley and predicted the return of Halley's Comet in late 1758. The exact time of the comet's return required the consideration of perturbations to its orbit caused by planets in the Solar System such as Jupiter, which Clairaut and his two colleagues Jérôme Lalande and Nicole-Reine Lepaute carried out more precisely than Halley, finding that the comet should appear in the constellation of Taurus. It is in searching in vain for the comet that Charles Messier found the Crab nebula, which he at first thought to be Halley's comet. After some observation, noticing that the object that he was observing was not moving across the sky, Messier concluded that the object was not a comet. Messier then realised the usefulness of compiling a catalogue of celestial objects of a cloudy nature, but fixed in the sky, to avoid incorrectly cataloguing them as comets.