We can take a scientific approach to the issue of mankind's capacity to calculate orbits accurately. Just look at the satellites they put up. They either fall back down, or require continual adjustments.
Sattelites that require adjustments need them because of drag with molecules from the atmosphere that they orbit in. The atmosphere does not end in a discrete barrier - it conintues to spread out thinning until it is gone. The sattelits orbit through a thin atmosphere. This is well known stuff - engineers have been dealing with it for quite sometime.
Mercury's orbit can be calculated with greater precision under Newton if one treats the sun as a large body rather than a point. The calculations are more tedious, however. According to Einstein himself, the simpler, less accurate Newtonian method was the one he "defeated".
According to Newtonian mechanics and Newton's law of gravitation, a planet which is revolving round the sun would describe an ellipse round the latter, or, more correctly, round the common centre of gravity of the sun and the planet. In such a system, the sun, or the common centre of gravity, lies in one of the foci of the orbital ellipse in such a manner that, in the course of a planet-year, the distance sun-planet grows from a minimum to a maximum, and then decreases again to a minimum.
You are becoming the king of linking and quoting things that do not support what you say. Not at any point in your link or your quote does Einstein day that he defeated a less accurate form of Newton's laws.
I personally know Newton's laws inside and out. I have studied them in their pure form and their modified form. All versions of Newton's laws of mechanics fail to properly calculate the proper orbit of Mercury. Please link the maths if you continue down this line of debate. It should be easy to find if you know it. Of course if you are simply making stuff up like you usually do (which I suspect) then you will find nothing.
Even so, anyone thinking even the simplest method gave poor results has a false impression. The margins-of-error involved are quite small.
The amount Newton's theory is off is significant enough to be measurable and was a major cause for concern.
As seen from Earth the precession of Mercury's orbit is measured to be 5600 seconds of arc per century (one second of arc=1/3600 degrees). Newton's equations, taking into account all the effects from the other planets (as well as a very slight deformation of the sun due to its rotation) and the fact that the Earth is not an inertial frame of reference, predicts a precession of 5557 seconds of arc per century. There is a discrepancy of 43 seconds of arc per century.
This discrepancy cannot be accounted for using Newton's formalism. Many ad-hoc fixes were devised (such as assuming there was a certain amount of dust between the Sun and Mercury) but none were consistent with other observations (for example, no evidence of dust was found when the region between Mercury and the Sun was carefully scrutinized). In contrast, Einstein was able to predict, without any adjustments whatsoever, that the orbit of Mercury should precess by an extra 43 seconds of arc per century should the General Theory of Relativity be correct.
It wasn't much but it should have been a measurable effect. And it was measured. And it was off by the amount predicted using GR.
None of the methods mentioned attempt to account for the gravitational interactions of all the other planets and moons.
CTD you have once again shown me that you have no clue what you are talking about. Both theories take into account the gravitational interactions of the other sources in our solar systems. Only one theory, GR, does so correctly. In fact, both theories predict a precession BECAUSE of the other planets in the solar system. However, Newton's theory does so incorrectly.
None account for magnetism either (it's stark verboten to even suggest magnetism might play a role). Perhaps that's why none actually match up perfectly with observations. Or maybe the calculations of the mass of the bodies might be off somewhere. Whoever figures out all the problems and obtains perfect accuracy might be up for a Nobel Prize, one supposes, provided their politics are acceptable to the committee...
Do you just type this stuff as you think of it? Please back up each statement in this paragraph with a paper please.