Snell-Descartes Law

Physical Law
Consider a ray of light crossing the threshold between two media.

Let its speed:
 * in medium $1$ be $v_1$
 * in medium $2$ be $v_2$.

Let it meet the threshold at:
 * an angle $\alpha_1$ from the vertical in medium 1,
 * an angle $\alpha_2$ from the vertical in medium 2.

Then the Snell-Descartes law states that:
 * $\dfrac {\sin \alpha_1} {v_1} = \dfrac {\sin \alpha_2} {v_2}$

Proof
The Snell-Descartes law can be derived from Fermat's Principle of Least Time as follows:

Let the ray of light travel from $A$ to $P$ in the medium $1$.

Then let it travel from $P$ to $B$ in medium $2$.


 * SnellsLaw.png

The total time $T$ required for that journey is:
 * $T = \dfrac {\sqrt{a^2 + x^2}} {v_1} + \dfrac {\sqrt{b^2 + \left({c - x}\right)^2}} {v_2}$

from the geometry of the above diagram.

From Fermat's Principle of Least Time, this time will be a minimum.

From Derivative at Maximum or Minimum, we need:
 * $\dfrac {\mathrm d T} {\mathrm d x} = 0$.

So:
 * $\dfrac x {v_1 \sqrt {a^2 + x^2} } = \dfrac {c - x} {v_2 \sqrt {b^2 + \left({c - x}\right)^2} }$

which leads directly to:
 * $\dfrac {\sin \alpha_1} {v_1} = \dfrac {\sin \alpha_2} {v_2}$

by definition of sine.

However, it had previously been discovered by several other scientists, including in 984 and  in 1602.

's discovery was made in 1621, while discovered it independently in 1637.

In France this law is called la loi de Descartes or loi de Snell-Descartes, while elsewhere it tends to be known as Snell's law, or Snell's Law of Refraction.