Coming out of the Dark Ages, man believed the Earth was flat, and that the heavens revolved around the Earth. Evidence that contradicted these assumptions was ignored until the weight of evidence became so heavy that the comfortable assumption was eroded. How could such a theory as the flat Earth even evolve? To man today, this theory seems laughable, but when it evolved, the edges of the horizon always seemed at the edge of a flat plane, so this was the newborn child's first conclusion. But did not the fact that the stars moved about the heavens in a manner that lined up with a solar system model not move him to question? The toddler assumes the world revolves around him, and is loath to let this go. The alternate explanation, that the heavens were dancing for his amusement, fit his mind-set. Did not the Sun rise when he felt refreshed, and set when he was weary? All to serve his needs, he had no doubt.
Elaborate mathematical descriptions of trajectories and orbits were drawn up in an age when man had not peeked beyond the Solar System with high powered telescopes floating above the atmosphere, and when slow motion video was unheard of. The only complete orbits known were those of the planets which hugged the Sun, and as the math was drawn up to fit these orbits, the orbits fit the math. The explanation for comets either fit the model or they didn't. When they fit the model they were assumed to have the orbit, when out of view, that the model dictated. If they didn't fit the model then they were dressed up in mathematical curves, parabolas or hyperbolas, which came close enough to let everyone go home at the end of the day feeling smug. As concepts tend to solidify as time passes, the young taking as absolutes what their elders preach, the Earth was now no longer flat, and the heavens no long revolved around the Earth, but most certainly all orbits were elliptical.
When slow motion video demonstrated that trajectories do not, in fact, mirror the downward side to the upward side, the facts did not change the precepts taught to the young. Why change a handy tool that works for everyday applications? Close enough, and change would require reprinting all those books! The fact that the downward side of the trajectory marries the forward thrust to the gravity drop was noted by those who think deeply about such matters, and is a known but not extensively taught fact. For most, the Earth is still flat, as they have not been told otherwise and are not inclined to question. Where the trajectory precept stands inviolate to most, the precepts of elliptical orbits have even fewer challenges. Man sees the dance of stars, but there are so many variables that come into play, about which he is uncertain at best, that the arrows of doubt seldom get launched. Yet the arrows exists.
Where orbits are snug about their center of gravity, there is little contradiction between these and what mankind calls their laws of gravity and motion. These are not laws, of course, but elaborate descriptions of what they observe. The flaws in the laws, however, were always present. If gravity diminishes with distance, but distance is attained with speed, then an object in a snug elliptical orbit seems to logically be adhering to the laws of gravity and motion. Speed up during the approach, sling past, and the speed carries the body outward where the diminishing gravity pull slows the body down so that its curve sidewards takes predominance. The theory fit what man observed, and thus was not questioned until his powers of observation increased. Tiny comets, seen by man in the past only when they gave their brilliant displays while going around the Sun, have only recently been observed in great detail during this passage.
Repeating comets are not slinging past, as in a passing body. They are in orbit, doing the better part of a circle about the Sun. Unlike the planets, whose center of gravity is just that, at the center, the comet does not behave as though the Sun is its center of gravity. The elliptical orbit of planets is such that if one were to examine the distance from the Sun, the difference at any given point would be slight. It is more circular than not. Comets, however, are at the other extreme. They appear to be a fan, rather than an eye. For the laws of gravity and motion to fit, the comet must be increasing its speed as it leaves the Sun, thus explaining its increasing distance. However, careful studies have shown this not to be the case. The comet is going its fastest when closest to the Sun, and has slowed down when it begins to leave what is assumed to be its gravitational master. The slowing precedes the exit, thus throwing the smug assumptions of man into consternation.
In addition, the distance comets travel outward, and the curvature of their exit are now able to be examined where in the past they were an unknown. They go essentially straight away, not the curve anticipated. Thus the distance from the Sun wherein they would have to complete an elliptical curve is extreme, challenging the laws of gravity and motion. The distance where the elliptical curve would reinstate is too far, and the curve during the straight away too slight. Rather than deal with this new information, the majority of scientists prefer their comfort factor over new knowledge. Change is resisted, and for many the Earth is still flat.