# When was the mathematical relation between Snell's law and Fermat time law first recognized?

In the wikipedia article about Snell's law of indexes of refraction there is a demonstration that shows the interconnection with Fermat's law of least time.

The strategy of that demonstration:
The overall travel time is a function of the variation of the path. The aspect that is to be compared is not the travel time itself, but the travel time derivative with respect to the path variation.

There is medium 1 and medium 2, and there is the interface between the two media. In the general case that interface is not at right angles to the line connecting the start point and the end point.

There is the travel time $$t_1$$ for the path from start point to the interface, and $$t_2$$ for the path from the interface to the end point.

$$t_1$$ has an individual minimum, and $$t_2$$ has an individual minimum. The respective minima do not coincide, so there is an interval in the variation space where $$t_1$$ is still descending whereas $$t_2$$ is alraady ascending again (or vice versa). The global minimum of the travel time occurs at the crossover point of one still descending and the other already ascending. At that crossover point the two derivatives match.

In physics derivations are bidirectional.

(There is a general discussion of derivation in physics by stackexchange contributor knzhou, writing: "in physics, you can often run derivations in both directions: you can use X to derive Y, and also Y to derive X. That isn't circular reasoning, because the real support for X (or Y) isn't that it can be derived from Y (or X), but that it is supported by some experimental data D. This two-way derivation then tells you that if you have data D supporting X (or Y), then it also supports Y (or X)."

The thrust of my question is the following:
Snell's law describes refraction as something that occurs locally. Light is propagating, and when it encounters an interface to a medium with a different index of refraction the direction of propagation changes right there, right at that instant in time.

Fermat's law of least time obtains the change of direction as a consequence of a global criterium; the global travel time.

So: at first sight the two approaches appear to be in some way the opposite of each other.

The derivation shows that the the two ways to obtain the angle of refraction are mathematically equivalent. So: the point is moot.

Was Fermat already aware of it? Huygens?

I'm guessing Newton was not aware of it. Newton favored a theory of light where propagation of light is thought of as motion of particles. My understanding is that Newton's theory of refraction required light to be faster in glass than in air.

(In general, theory of refraction is about ratios. A refraction experiment finds a ratio. It wasn't until the time of Fizeau that the technology was available to measure actual speed of light.)

(It could be, I haven't verified, that Fermat's law can be restated as a law of most time, under the newtonian assumption that in glass light propagates faster. That would mean that distinction between Fermat least time and Fermat most time is moot.)

With the above context in place:
It seems to me that awareness of the mathematical connection between Snell's law and Fermat's time law makes a lot of difference. So I think it's higly relevant to know when it was first recognized.

• It is hard to understand what the question is. Fermat himself derived Snell's law from the least time principle, and he used adequality, a precursor of taking derivatives, to derive it. Descartes, Huygens, Newton, etc., knew of it, Bernoulli even used it to solve the brachistochrone, but a mathematical equivalence does not resolve the physical question about the nature of light, on which there was disagreement. On debates over variational principles and where light travels faster see Lamborn's thesis. Mar 28, 2021 at 8:33
• @Conifold You are correct. I had failed to appreciate the following: the experimental data were angles of refraction. So for Fermat to show that the time law satisfies the experimental data he had to demonstrate that Snell's law and the time law are mathematically equivalent. The theory of refraction proposed by Huygens was the first to cast refraction as a wave propagation effect. The context of my post was/is that in modern textbooks this mathematical equivalence isn't mentioned (to my knowledge.) In other words: it turns out my point isn't about history. Mar 28, 2021 at 8:59
• @Conifold Thank you for the link to Lamborn's thesis. Lamborn offers that Maupertuis' proposal was not a mathematically coherent body of work. So I gather: apart from whether Maupertuis had priority it was a dead end anyway. The first coherent formulation of an action concept, it appears, was by William Rowan Hamilton. (Some claim Lagrange, but I gather Lagrange used d'Alembert's principle as starting point for his Lagrangian mechanics, hence no involvement of calculus of variations.) Mar 28, 2021 at 13:47