# Fermat's principle

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In optics, Fermat's principle or the principle of least time is the principle that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light.[1] However, this version of the principle is not general; a more modern statement of the principle is that rays of light traverse the path of stationary, not minimal, time.

Fermat's principle can be used to describe the properties of light rays reflected off mirrors, refracted through different media, or undergoing total internal reflection. It can be deduced from Huygens' principle, and can be used to derive Snell's law of refraction and the law of reflection.

## Contents

### Modern version

The historical form proposed by French mathematician Pierre de Fermat is incomplete. The modern version of Fermat's principle states that the optical path length must be stationary, which means that it can be either minimal, maximal or a point of inflection (a saddle point). Minima occur when a wave passes from one medium into another (refraction) and in the reflection of light from a planar mirror. Maxima occur in gravitational lensing. A point of inflection describes the path light takes when it is reflected off an elliptical mirrored surface.

### History

Hero of Alexandria (Heron) (c. 60) described a principle of reflection, which stated that a ray of light that goes from point A to point B, suffering any number of reflections on flat mirrors, in the same medium, has a smaller path length than any nearby path.[2]

Ibn al-Haytham (Alhacen), in his Book of Optics (1021), expanded the principle to both reflection and refraction, and expressed an early version of the principle of least time. His experiments were based on earlier works on refraction carried out by the Greek scientist Ptolemy[3]

The generalized principle of least time in its modern form was stated by Fermat in a letter dated January 1, 1662, to Cureau de la Chambre.[4] It was met with objections made in May 1662 by Claude Clerselier, an expert in optics and leading spokesman for the Cartesians at that time. Amongst his objections, Clerselier states:

... Fermat's principle can not be the cause, for otherwise we would be attributing knowledge to nature: and here, by nature, we understand only that order and lawfulness in the world, such as it is, which acts without foreknowledge, without choice, but by a necessary determination.