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Universal Gravitation
Dependence of force of gravity on the masses of the bodies  
Imagine taking two pieces of the same type of metal, one having twice the volume of the other and hence twice the mass.  
We know from everyday experience that the piece with twice the mass will be twice as heavy as the other.  
Conclusion: the force of gravity acting on a mass near the earth is proportional to its mass.  
   
Newton took this logic to apply to the mass of the earth (or any similar body) too.  
In other words, imagine the mass of the earth to be increased (but keeping constant radius), he assumed that would increase the force proportionally too.  
Hence his suggestion that the force is proportional to the masses of both the bodies concerned (and therefore to the product of the masses).  
   
Dependence of force of gravity on the distance between the bodies  
Consider a large mass, for example the sun, being orbited by smaller masses, for example planets, as shown below.  
How does the force on a given body depend of the radius of the orbit (assumed for simplicity to be a circle).  
   
   
Imagine the sun to be the source of an influence "spreading out" through the space around it.  
At radius r this influence is distributed over the area of a sphere of radius r.  
Surface area of a sphere of radius r is given by  
 
and surface area of a sphere of radius 2r is  
 
Clearly, because of the squared term  
 
so we might reasonably assume that the strength of the "influence" at radius 2r will be 4 times less than that at radius r.  
   
In conclusion, the strength of the force of gravity might be expected to decrease with the square of the distance between the two bodies.  
Hence Newton's suggestion of an inverse square law for gravity.  
   
NB  
When asked how he came to formulate his law of gravity, a humble Newton answered that he reached his conclusions by simply "thinking about it continuously"... so, don't be discouraged, you don't have to be a genius... you just have to think about something continuously and you'll get the answers... maybe!  
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