Intuition about why gravity is inversely proportional to exactly square of distance between objects
What is the intuition behind why gravity is inversely proportional to exactly square of distance between two object and not cube or not some multiplier. Basically how Newton came up that its exactly square of distance? How it was validated it is in fact square of distance ?
The harder part of the problem was that tricky "distance between two objects" as applied to non-point objects. Their mass only appears to be effectively at the geometric centre for bodies composed of concentric shells, each of uniform density. Luckily, gravity in massive bodies generally causes that distribution of mass.
Imagine "gravity" spreading out in a sphere, like light from a bulb.
For each doubling of the distance, the sphere has four times the area. The surface area of the sphere is proportional to the square of the radius. If the same gravity is stretched over that sphere, the force of gravity would be inversely proportional to the square of the radius.
This gives sufficient intuition for most physicists of the time to believe that gravity was probably inverse square. Newton's real genius was that he was able to prove using mathematics that a planet moving in an inverse square law would obey Kepler's laws, something that his contemporaries had failed to do.
A well known story, related by de Moivre:
In 1684 Dr Halley came to visit [Newton] at Cambridge. After they had been some time together, the Doctor asked him what he thought the curve would be that would be described by the planets supposing the force of attraction towards the sun to be reciprocal to the square of their distance from it. Sir Isaac replied immediately that it would be an ellipse. The Doctor, struck with joy and amazement, asked him how he knew it. Why, saith he, I have calculated it.
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The inverse square law seems very intuitive to me for Newtonian gravity as mentioned above. It doesn't seem quite so intuitive to me when looking at "Einsteinian Gravity". I asked a question a long time ago in one of the Stack Exchanges about deriving Newton from General Relativity and a brilliant person showed me how this is done (but it's not easy). I don't think you can look at this geometrically with regards to General Relativity since mass determines the geometry.