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*The second "classical" test of general relativity was the advance of the perihelion of the orbit of Mercury.  There are many complex effects contributing to this, including gravitational perturbations from other planets and the effect of the oblateness of the Sun.  These are hard to calculate accurately, but, by 1900 it was known quite accurately that there was an "anomalous" precession, that is, a precession beyond all other known effects, of 43 arc seconds per century.  This is a very tiny effect, but astronomical measurements were sufficiently accurate by that time to show it clearly.

*The second "classical" test of general relativity was the advance of the perihelion of the orbit of Mercury.  There are many complex effects contributing to this, including gravitational perturbations from other planets and the effect of the oblateness of the Sun.  These are hard to calculate accurately, but, by 1900 it was known quite accurately that there was an "anomalous" precession, that is, a precession beyond all other known effects, of 43 arc seconds per century.  This is a very tiny effect, but astronomical measurements were sufficiently accurate by that time to show it clearly.

:This created quite a problem&mdash;physicists by then were accustomed to having their theories check out very accurately.  One proposal that was made, by Simon Newcomb and Asaph Hall, was that the exponent of the radius in the gravitational formula wasn't exactly 2.  He showed that, by choosing an exponent of <math>2+\delta</math>, the precession, as a fraction of a full orbit per planet's year, is <math>\delta/2</math>.  By setting <math>\delta</math> to .000000157, that is, an exponent of 2.000000157, Newcomb was able to get a precession of .000000078 revolutions per Mercury year, or 43 arcseconds per Earth year.  Unfortunately, any such fudging of the exponent gives the same precession, per planetary year, for ''any'' orbiting object.  This would have given the Moon a very noticeable precession, which was not observed.  There may have been another effect working against Newcomb's theory.  There is sometimes "intellectual inertia," a reluctance to accept new theories.  This would have been particularly evident in the case of Newtonian gravitation, because the Newtonian formula, with its exponent of exactly 2, can be integrated exactly, in closed form.  In fact, the invention of calculus, and mathematical physics in general, was largely founded on Newton's exact closed-form solution for an inverse square force field, and the demonstration that it exactly matched Kepler's laws.  Mathematical physics would have had a very rocky start if this weren't the case.  Giving up the exact Newtonian formulation, with its exponent of exactly 2, did not come easily.

:This created quite a problem&mdash;physicists by then were accustomed to having their theories check out very accurately.  One proposal that was made, by Simon Newcomb and Asaph Hall, was that the exponent of the radius in the gravitational formula wasn't exactly 2.  He showed that, by choosing an exponent of <math>2+\delta</math>, the precession, as a fraction of a full orbit per planet's year, is <math>\delta/2</math>.  By setting <math>\delta</math> to .000000157, that is, an exponent of 2.000000157, Newcomb was able to get a precession of .000000078 revolutions per Mercury year, or 43 arcseconds per Earth year.  The primary resistance to this approach came from mathematicians unable to do the integration without an exponent of precisely 2, and they insisted, incorrectly, that was impossible for the exponent to be slightly different from 2.  Due to this desire for mathematical elegance rather than objective observation-based science, Newcomb's approach was not pursued.

:While Newcomb's theory, and general relativity, don't lead to closed-form solutions, both theories can be solved numerically to as much precision as one desires.

:While Newcomb's theory, and general relativity, don't lead to closed-form solutions, both theories can be solved numerically to as much precision as one desires.

:General relativity predicts this extra precession, in the amount of <math>3{}v^2/c^2</math> revolutions per planet's "year", where <math>v</math> is the planet's average orbital speed.<ref>That is a simple approximation, designed to relate the precession to the planet's speed relative to the speed of light.  A more accurate approximation is <math>\frac{3GM}{c^2 a(1-e^2)}</math>, where a is the semi-major axis and e is the eccentricity.</ref>  More accurate, modern measurements of Mercury's precession differ slightly from the calculated combined effects of planetary perturbation, equinoctial precession, solar oblateness, and general relativity.  The reason for this is not known.

:Increasingly precise measurements of the precession demonstrate that it conflicts with General Relativity, despite claims of relativists for decades that it predicted the precession accurately in the amount of <math>3{}v^2/c^2</math> revolutions per planet's "year", where <math>v</math> is the planet's average orbital speed.<ref>That is a simple approximation, designed to relate the precession to the planet's speed relative to the speed of light.  A more accurate approximation is <math>\frac{3GM}{c^2 a(1-e^2)}</math>, where a is the semi-major axis and e is the eccentricity.</ref>  The conflict is greater than the margin of error, and many relativists avoid the discrepancy rather than address it.

:The following table show some approximate parameters for the planets.  Note that Mercury has the smallest orbit, the fastest speed, and the highest gravitational pull.  Precession of planets other than Mercury is extremely hard to measure, but measurements of the actual anomalous precessions are in good agreement with the last column of the table.<ref>http://www.mathpages.com/rr/s6-02/6-02.htm</ref>

:The following table show some approximate parameters for the planets.  Note that Mercury has the smallest orbit, the fastest speed, and the highest gravitational pull.  Precession of planets other than Mercury is extremely hard to measure, but measurements of the actual anomalous precessions are in good agreement with the last column of the table.<ref>http://www.mathpages.com/rr/s6-02/6-02.htm</ref>