# Fermat's Principle of Least Time

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## Physical Law

### Original formulation

The path taken by a ray of light from one point to another is the one that can be traversed in the least time.

### Modern version

The optical path length must be stationary, that is, be either a minimum, maximum or a point of inflection.

## Proof

This follows from the Huygens-Fresnel Principle.

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## Also see

## Source of Name

This entry was named for Pierre de Fermat.

## Historical Note

While Fermat's Principle of Least Time is named for Pierre de Fermat, it has been understood from antiquity.

Its first appearance seems to have been when it was applied by Heron of Alexandria when he formulated his Principle of Reflection.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{IV}$: The Prince of Amateurs - 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 $\text {A}.7$: Heron (first century A.D.) - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.13$: Fermat ($\text {1601}$ – $\text {1665}$)