Uniqueness of refractive index Let $A$ and $B$ be two media.  Let $\alpha$ be the angle of incidence of light on medium $B$ and let $\beta$ be the angle of refraction. Let  $_{_{A}}n_{{_B}}$ be the refractive index of $B$ with respect to $A$.
We know that $_{_{A}}n_{{_B}} = \dfrac{\sin\alpha}{\sin\beta}$
$_{_{A}}n_{{_B}}$ can also be expressed in terms of the speed of light in $A$ and $B$.
Let the speed of light in $A$ be $c_{_{A}}$ and in $B$ be $c_{_{B}}$
So, $_{_{A}}n_{{_B}} = \dfrac {c_{_{A}}}{c_{_{B}}}$
As the refractive index is a constant, this means that both the values of the refractive index must be the same.
This means that for any two media, the ratio of the sines of the angles of incidence and refraction will be equal to the ratio of the speed of light in both the media.
Is there some proof other than experimental proof for this?
How was it discovered?
 A: The key to this formula is Fermat's principle of least time. That is, the path given by 
$$n=\frac{\sin\alpha}{\sin\beta}$$
ensures that the light travel time is minimized.
Fermat's principle is itself a special case of the principle of stationary action, which has widespread application throughout physics. 
Fermat's principle is of historical significance as the first example of the principle of stationary action. Fermat himself couldn't fully justify why the principle of least time had to apply in the refraction of light, but it seemed reasonable at the time to assume that it did so. It wasn't until much later that Fermat's principle was given a solid theoretical basis as a case of the principle of stationary action. 

You can read about more about the historical details in the linked Wikipedia articles. But I'd like to mention a couple of them here because this issue was rather important in the history of physics.
Ptolemy stated that the refraction law is (in modern notation)
$$n=\frac{\alpha}{\beta}$$
He claimed that he had empirically verified this theory, but it seems pretty clear that he fudged the data to fit his theory. To be fair, Ptolemy didn't have access to a modern optics lab, and it's a reasonable approximation for small angles, since $\sin\theta \approx \theta$ when $\theta$ is small. (That's the well-known small angle approximation which is often useful, eg in approximating the period of a simple pendulum.)
About 800 years later, the Persian mathematician and physicist Ibn Sahl discovered the true law of refraction. Eventually, about 500 years later, this became known in Europe as Snell's law.
