Pierre de Fermat Derived Snell's Law from his Principle of Least Time

After I wrote my post Pierre de Fermat's Principle of Least Time, I received two interesting comments which merit a detailed response.

The first comment came from Brent Shadboldt (A Universe without Relativity, brentshadbolt.substack.com), who wanted me to clarify the differing positions of René Descartes (1596-1650) and Pierre de Fermat (1607-1665) with respect to refraction, especially in the light that Christiaan Huygens (1629-1695) supported Fermat’s position:

I’m interested in your mention of Huygens’ quote, where he appears to side with Fermat’s principle of least time over Descartes’ view. Huygens emphasizes that light chooses the path of least time, not necessarily least distance. However, if Huygens understood that least time implies light travels at varying speeds in different media, why would he then disagree with Descartes’ assertion that light slows down in denser materials? It seems like a contradiction. Could you please clarify Huygens’ position on this apparent discrepancy?

The answer is that Descartes argued that the resistance to the passage of light was lower in denser materials, not higher. Here is the relevant passage in his Dioptrics¹, published in 1637:

You will cease to this find strange, if you remember the nature that I attributed to light, when I said that it is nothing other than a certain movement or an action, received in a very subtle material that fills the pores of other bodies; and if you consider that, as a ball loses more of its motion in impacting a soft body than a hard one, and that it rolls less easily on carpet than on a smooth table, thus the action of this subtle material can be much more impeded by the parts of air, which are soft and disconnected and do not offer much resistance, than by those of water which give more resistance; and still more by those of glass or crystal: such that to the extent that the small pieces of a transparent body are harder and firmer, so much the more easily will they allow light to pass, for the light does not have to drive them out of their places as does a ball in displacing the parts of water in order to find passage between them. [p.21 (Descartes p.23)]

So in this paragraph, we can read that Descartes assumed that the small parts of a hard substance were themselves hard, while the small parts of a soft substance were themselves soft, and that the former would allow light to pass more easily, while the latter would hinder the passage of light. Note that I refer to the passage of light, as opposed to the speed of light, because Descartes initially thought that light passed instantly from emitter to receptor. See my post Galileo Galilei, Isaac Beeckman and René Descartes Discuss the Speed of Light.

Fermat took the opposite position. First, he always assumed that the speed of light was finite. Second, he assumed that the speed of light was lower in denser media. In a letter² dated August 1657 to Marin Cureau de la Chambre (1594-1669), Fermat introduces the Principle of Least Time. Therein he argues as follows:

But it is necessary to move on and find the reason for refraction in our common principle, which is that nature always acts along the shortest and easiest paths. It seems at first that this could not succeed and that you yourself have made an objection which appears invincible. […] But we can easily extricate ourselves from this difficulty by assuming, with you and with all those who have treated this matter, that the resistance of the media is different, and that there is always a certain ratio or proportion between those two resistances, when the two media have particular consistencies and are internally uniform. [p.74]

Do not be shocked by what I say about resistance, after you have decided that the movement of light is instantaneous and that refraction is caused by the natural antipathy which exists between light and matter. For whether you grant me that the instantaneous movement of light can be contested and that your proof is not entirely demonstrative, or whether it is necessary to stand by your decision, viz., that light avoids the abundance of matter which is its enemy, I find, even in the latter case, that since light avoids matter and that one only avoids that which causes pain and resists, one can, without departing from your conception, establish resistance where you establish avoidance or aversion. [p.75]

Fermat concludes as follows:

I will first deduce: that the perpendicular ray is not bent at all; that after light bends when passing from the first medium, it continues in the second without further changing the slope it has taken; that the bent ray sometimes approaches the perpendicular, and at other times departs from it, depending on whether it passes from a rarer to a denser medium or the contrary; and in a word, that this opinion is exactly in agreement with all appearances. [p.77]

The second comment was from Michael Clarage (Michael’s Newsletter, michaelclarage.substack.com), who wanted me to illustrate how Fermat had been able to prove Snell’s law, named after Willebrord Snellius [Willebrord Snel van Royen] (1580-1626), although it has been known as early as Ibn Sahl (940-1000): the sines of the angles of incidence and refraction are in inverse relation with the refractive indices of the two media and in direct relation with the speeds of light in the two media.

I was hoping for the proof that AB-BC is shorter time.

Fermat ultimately, in January 1662³, directly derived the sine law of refraction from his principle of least time and his hypothesis that light moves more slowly in denser media. The discussion uses the following diagram:

CD-DI is the path of least time

Let ACBI be a circle whose diameter AFDB separates two different media, with the less dense being on the side ACB, and the more dense being on the side AIB.

Let D be the center of the circle, and let CD be an incident ray falling upon the center from the given point C; we seek to determine the refracted ray DI, or the point I through which the ray passes after refraction.

Drop perpendiculars CF and IH onto the diameter. Since point C is given, along with the diameter AB and the center D, the point F and the line FD are also taken as given. Let the ratio of the media—namely, the ratio of the resistance of the denser medium to the resistance of the rarer medium—be equal to the ratio between the given straight line DF and another given line m external to the figure. Then, we will have m<DF, granted that the resistance of the rarer medium be less than the resistance of the denser medium—an axiom that is beyond natural.

By means of the lines m and DF, we now have to measure the motions along lines CD and DI; we will thus be able to represent the entirety of the motion along these two lines by the sum of two products: CD·m + DI·DF.

Thus the question comes down to dividing the diameter AB at a point H, such that if a perpendicular HI be erected at this point, and DI be joined, the area CD·m + DI·DF will be a minimum. [pp.150-151]

The sum CD·m + DI·DF corresponds to the total time for a light ray to reach from C to I, with the assumption that in the rarer medium, the ray passes from C to D, and in the denser medium, from D to I. Using his method of determining maxima and minima, his precursor to the infinitesimal calculus, Fermat determines that the sum is at a minimum when the length DH is exactly m. He concludes:

And thus the ray, passing from a rarer to a denser medium, will bend away from the side and toward the perpendicular: which agrees absolutely and without exception with the theorem discovered by Descartes. [p.153]

As an aside, here is my explanation of Fermat’s method: it is presupposed that when a curve reaches a maximum or a minimum at a point a, right around a, there can be no other maximum or minimum. So, he compares the values of the curve at two points: point a and point a+e, where a+e can be considered to be one indivisible away from a, and from there can compute what should be the value of a.

As I wrote in my post Bonaventura Cavalieri Introduces the Indivisibles, the indivisibles were introduced by Bonaventura Cavalieri (1598-1647) in his book Geometria indivisibilibus continuorum nova quadam ratione promota [A geometry of continua by way of indivisibles advanced in a new manner], published in 1635, and translated into Italian⁴ in 1966. And Fermat very quickly learnt of Cavalieri’s work:

In the first period after the publication of Geometria, the appreciation and interest of a French gentleman, a geometry enthusiast, Jean de Beaugrand, was noted. With Beaugrand, Cavalieri had a few hours of conversation in Bologna in the autumn of 1635. He writes to Galilei (on 24 December 1635): “I have had such a dear opportunity to have communication with those master mathematicians of France, given the scarcity there is here in Italy”; And to Giannantonio Rocca, a gentleman from Reggio Emilia, one of Cavalieri’s first and best students at the Studio di Bologna, he let it be known that he had made a gift of the Geometria to Beaugrand, who was in contact with “a certain Senator of Toulouse” (an important document, because it leads one to reasonably conjecture that around 1636 the “Senator of Toulouse”, Pierre Fermat, had certain and detailed knowledge of Cavalieri's method).⁵

So the final question is: Given that Descartes’s reasoning was completely fallacious, how did he come up with what is known in French as «la loi de Descartes» (“Descartes’s law”)? I think it is quite plausible that Descartes had access to some of Snellius’s unpublished work, invented a plausible explanation, and then published the work as his own. This would not be unheard of for Descartes. In my post Losing Marin Mersenne's Correspondence, I already hinted at the possibility that Descartes simply copied some of Isaac Beeckman’s (1588-1637) work. This subject merits further research.

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1. René Descartes. Dioptrics. In Pierre de Fermat. The Battle for Light: Fermat vs. Descartes. A Sourcebook on Pierre de Fermat’s Principle of Least Time. Translated and edited by Jason Ross. 2024, pp.3-20.

2. Fermat to de la Chambre, August 1657. In The Battle for Light, pp.71-79.

3. Analysis for Refraction, January 1, 1662. In The Battle for Light, pp.150-157.

4. Bonaventura Cavalieri. Geometria degli Indivisibile. A cura di Lucio Lombardo-Radice. Torino: Unione Tipografico-Editrice Torinese, 1966.

5. Segnaliamo, nel primo periodo dopo la pubblicazione della Geometria, l’apprezzamento e l’interesse di un gentiluomo francese, appassionato di geometria, Jean de Beaugrand. Con il Beaugrand il Cavalieri ha qualche ora di conversazione a Bologna nell’autunno del 1635. Scrive a Galilei (il 24 dicembre 1635): «ho avuto molto caro un’occasione tale per avere la communicazione con quei Sri matematici della Francia, stante la penuria che vi è qua in Italia»; e a Giannantonio Rocca, gentiluomo reggiano, uno dei primi e migliori allievi di Cavalieri allo Studio di Bologna, fa sapere di avere fatto omaggio della Geometria degli Indivisibili al Beaugrand, il quale è in relazione con «un tal Senatore di Tolosa» (documento importante, perché induce a congetturare ragionevolmente che attorno al 1636 il «Senatore di Tolosa», Pierre Fermat, avesse sicura e dettagliata notizia del metodo cavalieriano). [p.21]