How did Newton guess the inverse square law?

There has always been a great interest
in the motion of planets. By the 16th
century, a lot of data on the motion of

                                                                 

planets had been collected by many                                
astronomers. Based on these data
Johannes Kepler derived three laws,
which govern the motion of planets.
These are called Kepler’s laws. These are:
1. The orbit of a planet is an ellipse with
the Sun at one of the foci, as shown in
the figure given below. In this figure O
is the position of the Sun.
2. The line joining the planet and the Sun
sweep equal areas in equal intervals
of time. Thus, if the time of travel from
A to B is the same as that from C to D,
then the areas OAB and OCD are
equal.
3. The cube of the mean distance of a
planet from the Sun is proportional to
the square of its orbital period T. Or,
r3/T2 = constant.
It is important to note that Kepler
could not give a theory to explain
the motion of planets. It was Newton
who showed that the cause of the
planetary motion is the gravitational
force that the Sun exerts on them. Newton
used the third law
of Kepler to
calculate the
gravitational force
of attraction. The
gravitational force
of the earth is
weakened by distance. A simple argument
goes like this. We can assume that the
planetary orbits are circular. Suppose the                
orbital velocity is v and the radius of the
orbit is r. Then the force acting on an
orbiting planet is given by F v2/r.
If T denotes the period, then v = 2πr/T,
so that v2 r 2/T2.
We can rewrite this as v2 (1/r) ×
( r3/T2). Since r3/T2 is constant by Kepler’s
third law, we have v2 1/r. Combining
this with F v2/ r, we get, F 1/ r2.