Joseph Priestly indirectly deduced the inverse square law for the force between electric charges by analysing the electric force on a charge object inside a hollow conducting object. He knew already that a charged body inside a conducting body experienced no force. His thought experiment took place inside a sphere.
The ratio of the two areas above is
If the charge is uniformly distributed, the charge is proportional to the area. The particle experiences no force, so the force due to the charges on
and
cancel, meaning that the forces must be proportional to the charge and inversely proportional to the distance. The areasandare arbitrary, so the force on the particle will always be zero.
It is important to notice that the argument is not an experimental one and so not rigorous. That had to wait until Coulomb conducted his experiments.