So you can not just take the value of gravitational attraction at earths surface and apply inverse square law out in to space.

The above seems to be the issue you're trying to raise, but that issue seems to be incomplete to me as it's not clear where you're locating the effective point source origin of the force that should follow the inverse square law.

There is not one point source every particle attracts every other particle from where it actually is in reality.

And the total vector sum of these forces mean that a body's acceleration turns out to be the same as if it had been acted on by a single force towards earths centre.

However as i said previously

Newton did not allow for the variation in the amount of cancellation due to the vector nature of gravitational attraction when at different distances in space from a mass.

It would indeed be incorrect to use the inverse square law from an origin on the surface of the Earth, but is that what you're actually getting at? The measured value at the Earth's surface does seem to follow the inverse square law from an imaginary point origin at the Earth's center.

I know that newtons theory takes inverse square law from the centre of the earth and it is the value attraction at earths surface that is used, sorry for not making that clearer.

So to return to newtons error, to give a basic example.

If you had a standard 1 kg mass on a frictionless surface and applied a force of 3 newtons in one direction and also applied a force of 4 newtons at 90 degrees to the other force on the body you would have a resulting acceleration of 5 m/s squared just like a 3-4-5 triangle.

The total force that is applied to the 1 kg mass is however 7 newtons and if that force was applied at a angle of anything less than 90 degrees the amount of acceleration of the mass would be greater as the amount of cancellation of the vector sum would be less.

So if you then consider a body on the earths surface the mass that is attracting it to the centre it is mostly attracting at a vector angle, there is only a very small amount directly in line with a body's direction of acceleration.

The question that then arises is what is the total force that is acting on a body at earths surface as we already know what the resulting acceleration force is, so at what angle is the average cone of attraction and how much cancellation in the vector sum at earths surface.

And when you put space between a body and earths mass lets say one earths radius from earths surface so two radius's from the centre newtons method would bring the force of attraction to one quarter of what it is at the surface with no allowance for the change in the slight lessening of the average angle of attraction.

But when you then bring the physical reality's in that earth is not a homogeneous sphere and you have to allow for the fact half of earths volume is within the last 1315 kilometres of earths radius but the earths core is thought to be a lot denser than the rest of the earth, So lots of variable confusing the issues.

And when you then consider that the total force will only apply close to a its total figure many earth radius's away when the angle becomes much more acute.

As it will have been diluted down hugely by the inverse square of the distance, so may be not surprising it is difficult detect given all the other dynamics of attraction going on in space.

At the core of what i am pointing out are very basic fundamentals of physics although given that gravity is a very weak force i understand how difficult this total force will be to detect.

However D longs paper about experiments to measure gravitational attraction from which the gravitational constant is derived

http://www.springerlink.com/content/u5j4724173336606/ is about the fact that the majority of results It seems from experimental results show inverse square law violation also.

It appears when looking at the data of experiments made since 1894 they show a dependence on mass separation.

And i say again Einstein included the gravitational constant in his calculations and the fact that G [big G ] is giving results closer to what is observed is due to the fact it came to a higher value for gravitational attraction by accident given the fact these experiments are using high density mass so getting more concentrated distance for the inverse square and more acute angles of attraction.