# 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 $2$.

The total time $T$ required for that journey is:

- $T = \dfrac {\sqrt {a^2 + x^2} } {v_1} + \dfrac {\sqrt {b^2 + \paren {c - x}^2} } {v_2}$

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:

- $\dfrac {\d T} {\d x} = 0$.

So:

- $\dfrac x {v_1 \sqrt {a^2 + x^2} } = \dfrac {c - x} {v_2 \sqrt {b^2 + \paren {c - x}^2} }$

which leads directly to:

- $\dfrac {\sin \alpha_1} {v_1} = \dfrac {\sin \alpha_2} {v_2}$

by definition of sine.

$\blacksquare$

## Also presented as

This law can also be seen expressed as:

- $\dfrac {\sin \alpha_1} {\sin \alpha_2} = \dfrac {v_1} {v_2}$

## Source of Name

This entry was named for Willebrord van Royen Snell and René Descartes.

## Historical Note

Willebrord van Royen Snell made his discovery in $1621$.

René Descartes was first to publish it, which he did in $1637$. Snell's own work did not appear until $1703$ when Christiaan Huygens included it in his *Dioptrica*.

However, it appears that Descartes' own statement was stated incorrectly as $\dfrac {\sin \alpha_1} {\sin \alpha_2} = \dfrac {v_2} {v_1}$, and proved inadequately.

It may well indeed be that Descartes himself learned it from Snell, who now generally holds the credit for its discovery.

It needs to be noted that this law had previously been discovered by several other scientists, including Ibn Sahl in $984$ and Thomas Harriot in $1602$.

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**.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 6$: The Brachistochrone. Fermat and the Bernoullis - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.13$: Fermat ($\text {1601}$ – $\text {1665}$) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.22$: Bernoulli's Solution of the Brachistochrone Problem