Galileo "demonstrated" the phenomenon of constant acceleration using a water clock and a ball rolling down an inclined groove. In my experience, even with modern equipment it is difficult to obtain decent data on this sort of phenomenon; and even these data are typically not consistent with a true state of constant acceleration! There is no doubt that Galileo was quite aware of these flaws in his description; he was also quite happy to consign them to the realm of the "non-ideal" - i.e. the deviations from his predictions were due to imperfections in the ramp and the disturbance of the motion by the presence of air. Galileo argued that the results of a falling-body experiment performed underwater would be a lot worse than those of his experiments in air, so that one merely needed to extrapolate to no medium at all (i.e. perfect vacuum) to obtain results in perfect agreement with his predictions!
This overtly Platonic idealism was not new; but Galileo had hit upon a "good" approximation - one which actually did work better and better as the circumstances got closer and closer to a well-defined ideal case. The corrections could be regarded as negligible perturbations upon an "essentially correct" idealization, to be beaten into submission either by improvement of the apparatus or by laborious calculations.
Thus began what I call the "Perturbation Paradigm" of Physics. This simple prescription - find a nice simple model that does "pretty well" and then "fix up" its inadequacies with a series of corrections or "perturbations" - is so powerful that we Physicists use it on almost everything. The recent history of elementary particle physics gives a particularly poignant example of how a problem that was seemingly intractable by this perturbative method (and which promised for a while to lead us into genuinely new ways of thinking, which might have been nice for a change) was finally recast into a form that allowed application of the Perturbation Paradigm after all. I will suppress the urge to tell you about it now. But just wait!