How did Coulomb find the law of value of electric force between two point charges ?

Coulomb supposed the charge on a metallic sphere is $q$. If the sphere is put in contact with an identical uncharged sphere, the charge will spread over the two spheres. By symmetry the charge on each sphere will be $\frac{q}{2}$.

Repeating this process we can get charges $\frac{q}{2}, \frac{q}{4}$ etc.

Spheres are obtained with pairs of electric charges of $\frac{q}{2}, \frac{q}{4}, \frac{q}{8}, \ldots$ by repeating this process.

Coulomb varied the distance for a fixed pair of charges and measured the force for different separations. Then he gave relation,

$\mathrm{F} \propto \frac{1}{r^{2}} \quad \ldots$ $(1)$

He then varied the charges in pairs, keeping the distance fixed for each pair. Comparing forces for different pairs of charges at different distances, Coulomb gave the relation

$\mathrm{F} \propto q_{1} q_{2}$

... $(2)$

Thus, the electric force between two electric charges, $\mathrm{F} \propto \frac{q_{1} q_{2}}{r^{2}}$

$\therefore \mathrm{F}=k \frac{q_{1} q_{2}}{r^{2}}$ where $k$ is coulombian constant.