Fermat's Theory

When light or a body travels from one point to another in a straight line, it follows the path of shortest distance and time.

When light is reflected, it likewise travels along the path of shortest distance and briefest time.

The equality of the angles of incidence and reflection result from requiring a body to travel from one point to another along the path of shortest distance and briefest time that involves a reflection from a given plane.

For if the angles are equal, the sum of the two paths by which the ball travel and return is shorter in distance and briefer in time than any other sum of paths making unequal angles.

Hence, both the direct and reflected motion of light depends on a metaphysical law that Nature always acts in the simplest possible manner to produce its effects.

Whether a material body travels from one point to another freely or obstructed, Nature always leads it along the path of:

• shortest distance
• briefest time.

Principle Applied to Refraction

Two transparent media are separated by the plane of their common surface.

The light’s point of departure is within one medium.

Its point of arrival is in the other medium.

But the line joing the two points is not perpendicular to the surface separating the media.

Let us also assume that the light travels with different speeds in the different media.

Hence, the straight line joining the two points is still the path of shortest distance.

But is no longer the path of briefest time.

The travel time depends on the speeds of light in the 2 media. The path of briefest time should be:

• longer in length in the medium where the light moves faster
• shorter in the medium where light moves slower.

This happens when light passes from air into water.

The ray is bent such that:

• a longer path is taken in air
• a shorter path is taken in water

Light moves faster in air than in water.

The bent path taken by the light causes it to move from its departure point to its arrival point in the briefest possible time.

This path-of-least-time principle of Fermat accounts for light’s:

• refraction
• direct propagation and reflection

Fermat easily believed that light traveled faster and more easily in less dense media.

Who would assume that light moves faster and more easily in glass and water than in the air or a vacuum?

Several celebrated mathematicians also agreed with Fermat’s principle, particularly Leibniz, who gave the problem an elegant mathematical analysis.

He was charmed by the metaphysical principle as an example of the final causes to which he was so attached.

He considered it beyond doubt that light moves more quickly in air than in water or glass.

Descartes believed exactly the opposite, that light moves more quickly in denser media.

His reasoning was perhaps inadequate. But that failing does not stem from his assumption on the speed of light.

Every theory of refraction makes some assumption about the relative speed of light in different media.

• Some agree with Descartes
• Some agree with Fermat and Leibniz

If light moves faster in denser media, the entire theory of Fermat and Leibniz is destroyed.

• The bent path of light upon refraction would correspond neither to the path of shortest distance nor to the path of briefest time.
• A path that traveled longer in a medium of slower speed would definitely not arrive in the shortest possible time.

Mr. de Mairan wrote about reflection and refraction and described:

• the history of the dispute between Fermat and Descartes
• the confusion and inability to harmonize the law of refraction with the metaphysical principle.