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! 
