By comparing together the Accounts of the Motions of these Comets, 'tis apparent, their Orbits are dispos'd in no manner of Order; nor can they, as t… - Edmond Halley

Next to Ticho, came the Sagacious Kepler. He having the Advantage of Ticho's Labours and Observations, found out the true Physical System of the World, and vastly improv'd the Astronomical Science.
For he demonstrated that all the Planets perform their Revolutions in Elliptick Orbits, whose 'Plains pass thro' the Center of the Sun, observing this Law, That the Area's (of the Elliptick Sectors, taken at the Center of the Sun, which he proved to be in the common Focus of these Ellipses) are always proportional to the Times, in which the correspendent Elliptical Arches are describ'd. He discover'd also, That the Distances of the Planets from the Sun are in the Ratio [3:2] of the Periodical Times, or (which is all one) That the Cubes of the Distances are as the Squares of the Times. This great Astronomer had the Opportunity of observing Two Comets, one of which was a very remarkable one. And from the Observations of these (which afforded sufficient Indications of an Annual Parallax) he concluded, That the Comets mov'd freely thro' the Planetary Orbs, with a Motion not much different from a Rectilinear one; but of what Kind, he cou'd not then precisely determine.

[I]n the Year 1456, in the Summer time, a Comet was seen passing Retrograde between the Earth and the Sun, much after the same Manner: Which, tho' no Body made Observations upon it, yet from its Period, and the Manner of its Transit, I cannot think different from those I have just now mention'd. Hence I dare venture to foretell, That it will return again in the Year 1758. And, if it should then return, we shall have no Reason to doubt but the rest must return too: Therefore Astronomers have a large Field to exercise themselves in for many Ages, before they will be able to know the Number of these many and great Bodies revolving about the common Center of the Sun; and reduce their Motions to certain Rules.

Hitherto I have consider'd the Orbits of Comets as exactly Parabolick; upon which Supposition it wou'd follow, that Comets being impell'd towards the Sun by a Centripetal Force, descend as from Spaces infinitely distant, and by their Falls acquire such a Velocity, as that they may again run off into the remotest Parts of the Universe, moving upwards with such a perpetual Tendency, as never to return again to the Sun. But since they appear frequently enough, and since none of them can be found to move with an Hyperbolick Motion, or a Motion swifter than what the... Comet might acquire by its Gravity to the Sun, 'tis highly probable they rather move in very Excentrick Orbits, and make their Returns after long Periods of Time: For so their Number will be determinate, and, perhaps, not so very great. Besides, the Space between the Sun and the fix'd Stars is so immense, that there is Room enough for a Comet to revolve, tho' the Period of its Revolution be vastly long.