Editing Fermat's principle (section)

== History ==

If a ray follows a straight line, it obviously takes the path of least ''length''. [[Hero of Alexandria]], in his ''[[Catoptrics]]'' (1st century CE), showed that the ordinary [[specular reflection|law of reflection]] off a plane surface follows from the premise that the total ''length'' of the ray path is a minimum.<ref>[[#Sabra81|Sabra, 1981]], pp.{{nnbsp}}69–71. As the author notes, the law of reflection itself is found in Proposition&nbsp;{{serif|XIX}} of [[Euclid's Optics|Euclid's ''Optics'']].</ref> [[Ibn al-Haytham]], an 11th-century polymath later extended this principle to refraction, hence giving an early version of the Fermat's principle.<ref>{{Cite journal |last=Rashed |first=Roshdi |date=2019-04-01 |title=Fermat et le principe du moindre temps |journal= 	Comptes Rendus Mécanique |volume=347 |issue=4 |pages=357–364 |doi=10.1016/j.crme.2019.03.010 |bibcode=2019CRMec.347..357R |s2cid=145904123 |issn=1631-0721|doi-access=free |url=https://hal.archives-ouvertes.fr/hal-03485054/file/S163107211930052X.pdf }}</ref><ref>{{Cite book |last=Bensimon |first=David |url=https://books.google.com/books?id=YTVPEAAAQBAJ&dq=alhazen+fermat%27s+principle&pg=PT90 |title=The Unity of Science |date=2021-12-14 |publisher=CRC Press |isbn=978-1-000-51883-2 |language=en}}</ref><ref>{{Cite book |last1=Sanz |first1=Ángel S. |url=https://books.google.com/books?id=zR24BAAAQBAJ&dq=fermat%27s+principle+hero+ibn+haytham&pg=PA42 |title=A Trajectory Description of Quantum Processes. I. Fundamentals: A Bohmian Perspective |last2=Miret-Artés |first2=Salvador |date=2012-03-27 |publisher=Springer |isbn=978-3-642-18092-7 |language=en}}</ref>

=== Fermat vs. the Cartesians ===
[[File:Pierre de Fermat (F. Poilly).jpg|thumb|Pierre de Fermat {{nowrap|(1607{{r|katscher-2016}}{{hsp}}–1665)}}]]

In 1657, Pierre de Fermat received from [[Marin Cureau de la Chambre]] a copy of newly published treatise, in which La&nbsp;Chambre noted Hero's principle and complained that it did not work for refraction.<ref>[[#Sabra81|Sabra, 1981]], pp.{{nnbsp}}137–9; [[#Darr12|Darrigol, 2012]], p.{{nnbsp}}48.</ref>

Fermat replied that refraction might be brought into the same framework by supposing that light took the path of least ''resistance'', and that different media offered different resistances. His eventual solution, described in a letter to La&nbsp;Chambre dated 1&nbsp;January 1662, construed "resistance" as inversely proportional to speed, so that light took the path of least ''time''. That premise yielded the ordinary law of refraction, provided that light traveled more slowly in the optically denser medium.<ref>[[#Sabra81|Sabra, 1981]], pp.{{nnbsp}}139,{{px2}}143–7; [[#Darr12|Darrigol, 2012]], pp.{{nnbsp}}48–9 (where, in footnote&nbsp;21, "Descartes to&nbsp;..." obviously should be "Fermat to&nbsp;...").</ref><ref group=Note>[[Ibn al-Haytham]], writing in [[Cairo]] in the 2nd decade of the 11th century, also believed that light took the path of least resistance and that denser media offered more resistance, but he retained a more conventional notion of "resistance". If this notion was to explain refraction, it required the resistance to vary with direction in a manner that was hard to reconcile with reflection. Meanwhile [[Ibn Sahl (mathematician)|Ibn&nbsp;Sahl]] had already arrived at the correct law of refraction by a different method; but his law was not propagated&nbsp;([[#Mihas06|Mihas, 2006]], pp.{{nnbsp}}761–5; [[#Darr12|Darrigol, 2012]], pp.{{nnbsp}}20–21,{{px2}}41).<br/>The problem solved by Fermat is mathematically equivalent to the following: given two points in different media with different densities, minimize the ''density-weighted'' length of the path between the two points. In [[Leuven|Louvain]], in 1634 (by which time [[Willebrord Snellius]] had rediscovered Ibn Sahl's law, and Descartes had derived it but not yet published it), the [[Society of Jesus|Jesuit]] professor Wilhelm Boelmans gave a correct solution to this problem, and set its proof as an exercise for his Jesuit students&nbsp;([[#Zigg80|Ziggelaar, 1980]]).</ref>

Fermat's solution was a landmark in that it unified the then-known laws of geometrical optics under a ''[[variational principle]]'' or ''[[action (physics)|action principle]]'', setting the precedent for the [[principle of least action]] in classical mechanics and the corresponding principles in other fields (see ''[[History of variational principles in physics]]'').<ref>[[#Chaves16|Chaves, 2016]], chapters 14,{{tsp}}19.</ref> It was the more notable because it used the method of ''[[adequality]]'', which may be understood in retrospect as finding the point where the slope of an infinitesimally short [[chord (geometry)|chord]] is zero,<ref>[[#Sabra81|Sabra, 1981]], pp.{{nnbsp}}144–5.</ref> without the intermediate step of finding a general expression for the slope (the [[derivative]]).

It was also immediately controversial. The ordinary law of refraction was at that time attributed to [[René Descartes]] (d.{{nnbsp}}1650), who had tried to explain it by supposing that light was a force that propagated ''instantaneously'', or that light was analogous to a tennis ball that traveled ''faster'' in the denser medium,{{r|schuster-2000-261}}<ref>[[#Darr12|Darrigol, 2012]], pp.{{nnbsp}}41–2.</ref> either premise being inconsistent with Fermat's.&nbsp; Descartes' most prominent defender, [[Claude Clerselier]], criticized Fermat for apparently ascribing knowledge and intent to nature, and for failing to explain why nature should prefer to economize on time rather than distance. Clerselier wrote in part:
<blockquote>
1. The principle that you take as the basis of your demonstration, namely that nature always acts in the shortest and simplest ways, is merely a moral principle and not a physical one; it is not, and cannot be, the cause of any effect in nature&nbsp;.... For otherwise we would attribute knowledge to nature; but here, by "nature", we understand only this order and this law established in the world as it is, which acts without foresight, without choice, and by a necessary determination.

2. This same principle would make nature irresolute&nbsp;... For I ask you&nbsp;... when a ray of light must pass from a point in a rare medium to a point in a dense one, is there not reason for nature to hesitate if, by your principle, it must choose the straight line as soon as the bent one, since if the latter proves shorter in time, the former is shorter and simpler in length? Who will decide and who will pronounce?{{nnbsp}}{{r|clerselier-1662}}
</blockquote>

Fermat, being unaware of the mechanistic foundations of his own principle, was not well placed to defend it, except as a purely geometric and [[kinematics|kinematic]] proposition.{{r|smith-1959-651n|fermat-1662-clerselier}}&nbsp; The [[wave theory of light]], first proposed by [[Robert Hooke]] in the year of Fermat's death,<ref>[[#Darr12|Darrigol, 2012]], p.{{nnbsp}}53.</ref> and rapidly improved by [[Ignace-Gaston Pardies]]<ref>[[#Darr12|Darrigol, 2012]], pp.{{nnbsp}}60–64.</ref> and (especially) [[Christiaan Huygens]],<ref>[[#Darr12|Darrigol, 2012]], pp.{{nnbsp}}64–71; [[#ToL|Huygens, 1690, tr.&nbsp;Thompson]].</ref> contained the necessary foundations; but the recognition of this fact was surprisingly slow.

=== Huygens's oversight ===

[[File:Christiaan-huygens4.jpg|left|thumb|Christiaan Huygens {{nowrap|(1629–1695)}}]]

In 1678, Huygens proposed that every point reached by a luminous disturbance becomes a source of a spherical wave; the sum of these secondary waves determines the form of the wave at any subsequent time.<ref>Chr. Huygens, ''[[Treatise on Light|Traité de la Lumière]]'' (drafted 1678; published in Leyden by Van der Aa, 1690), translated by [[Silvanus P. Thompson]] as ''[[iarchive:treatiseonlight031310mbp|Treatise on Light]]'' (London: Macmillan, 1912; [http://www.gutenberg.org/ebooks/14725 Project Gutenberg edition], 2005), p.19.</ref> Huygens repeatedly referred to the envelope of his secondary wavefronts as the ''termination'' of the movement,<ref>[[#ToL|Huygens, 1690, tr.&nbsp;Thompson]], pp.{{nnbsp}}20,{{hsp}}24,{{hsp}}37,{{hsp}}51,{{hsp}}80,{{hsp}}108,{{hsp}}119,{{hsp}}122 (with various inflections of the word).</ref> meaning that the later wavefront was the outer boundary that the disturbance could reach in a given time,<ref>[[#ToL|Huygens, 1690, tr.&nbsp;Thompson]], top of p.{{nnbsp}}20.</ref> which was therefore the minimum time in which each point on the later wavefront could be reached. But he did not argue that the ''direction'' of minimum time was that from the secondary source to the point of tangency; instead, he deduced the ray direction from the extent of the common tangent surface corresponding to a given extent of the initial wavefront.<ref>Cf.{{tsp}} [[#ToL|Huygens, 1690, tr.&nbsp;Thompson]], {{nowrap|pp.{{tsp}}19–21,{{px2}}63–5}}.</ref> His only endorsement of Fermat's principle was limited in scope: having derived the law of ordinary refraction, for which the rays are normal to the wavefronts,<ref>[[#ToL|Huygens, 1690, tr.&nbsp;Thompson]], pp.{{nnbsp}}34–9.</ref> Huygens gave a geometric proof that a ray refracted according to this law takes the path of least time.<ref>[[#ToL|Huygens, 1690, tr.&nbsp;Thompson]], pp.{{nnbsp}}42–5.</ref> He would hardly have thought this necessary if he had known that the principle of least time followed ''directly'' from the same common-tangent construction by which he had deduced not only the law of ordinary refraction, but also the laws of rectilinear propagation and ordinary reflection (which were also known to follow from Fermat's principle), and a previously unknown law of [[birefringence|extraordinary refraction]] &ndash; the last by means of secondary wavefronts that were [[spheroid]]al rather than spherical, with the result that the rays were generally oblique to the wavefronts. It was as if Huygens had not noticed that his construction implied Fermat's principle, and even as if he thought he had found an exception to that principle. Manuscript evidence cited by Alan E.{{tsp}}Shapiro tends to confirm that Huygens believed the principle of least time to be invalid "in [[birefringence|double refraction]], where the rays are not normal to the wave fronts".<ref>[[#Shapiro73|Shapiro, 1973]], p.{{nnbsp}}229, note&nbsp;294 (Shapiro's words), citing Huygens' ''Oeuvres Complètes'', vol.{{nnbsp}}13 (ed.&nbsp;[[Diederik Korteweg|D.J.{{nnbsp}}Korteweg]], 1916), [https://www.dbnl.org/tekst/huyg003oeuv13_01/huyg003oeuv13_01_0053.php Quatrième Complément à la ''Dioptrique''], at p.{{nnbsp}}834, "Parte&nbsp;2<sup>da</sup>&nbsp;..." (in Latin, with annotations in French).</ref><ref group=Note>In the last chapter of his ''[[#ToL|Treatise]]'', Huygens determined the required shapes of image-forming surfaces, working from the premise that all parts of the wavefront must travel from the object point to the image point in ''equal'' times, and treating the rays as normal to the wavefronts. But he did not mention Fermat in this context.</ref>

Shapiro further reports that the only three authorities who accepted "Huygens' principle" in the 17th and 18th centuries, namely [[Philippe de La Hire]], [[Denis Papin]], and [[Gottfried Wilhelm Leibniz]], did so because it accounted for the extraordinary refraction of "[[Iceland spar|Iceland crystal]]" (calcite) in the same manner as the previously known laws of geometrical optics.<ref>[[#Shapiro73|Shapiro, 1973]], pp.{{nnbsp}}245–6,{{tsp}}252.</ref> But, for the time being, the corresponding extension of Fermat's principle went unnoticed.

=== Laplace, Young, Fresnel, and Lorentz ===

[[File:Laplace, Pierre-Simon, marquis de.jpg|thumb|Pierre-Simon Laplace {{nowrap|(1749–1827)}}]]

On 30 January 1809,{{r|laplace-1809}} [[Pierre-Simon Laplace]], reporting on the work of his protégé [[Étienne-Louis Malus]], claimed that the extraordinary refraction of calcite could be explained under the corpuscular theory of light with the aid of [[Maupertuis's principle]] of least action: that the integral of speed with respect to distance was a minimum. The corpuscular speed that satisfied this principle was proportional to the reciprocal of the ray speed given by the radius of Huygens' spheroid. Laplace continued:
<blockquote>
According to Huygens, the velocity of the extraordinary ray, in the crystal, is simply expressed by the radius of the spheroid; consequently his hypothesis ''does not agree'' with the principle of the least action: but ''it is remarkable'' that it agrees with the principle of Fermat, which is, that light passes, from a given point without the crystal, to a given point within it, in the least possible time; for it is easy to see that this principle coincides with that of the least action, if we invert the expression of the velocity.<ref>Translated by [[#Young1809|Young (1809)]], p.{{nnbsp}}341; Young's italics.</ref>
</blockquote>

[[File:Thomas Young (scientist).jpg|left|thumb|Thomas Young {{nowrap|(1773–1829)}}]]

Laplace's report was the subject of a wide-ranging rebuttal by [[Thomas Young (scientist)|Thomas Young]], who wrote in part:
<blockquote>
The principle of Fermat, although it was assumed by that mathematician on hypothetical, or even imaginary grounds, is in fact a fundamental law with respect to undulatory motion, and is {{sic|explicitly}} the basis of every determination in the Huygenian theory...&nbsp; Mr.&nbsp;Laplace seems to be unacquainted with this most essential principle of one of the two theories which he compares; for he says, that "it is remarkable" that the Huygenian law of extraordinary refraction agrees with the principle of Fermat; which he would scarcely have observed, if he had been aware that the law was an immediate consequence of the principle.<ref>[[#Young1809|Young, 1809]], p.{{nnbsp}}342.</ref>
</blockquote>
In fact Laplace ''was'' aware that Fermat's principle follows from Huygens' construction in the case of refraction from an isotropic medium to an anisotropic one; a geometric proof was contained in the long version of Laplace's report, printed in 1810.<ref>On the proof, see [[#Darr12|Darrigol, 2012]], p.{{nnbsp}}190. On the date of the reading (misprinted as 1808 in early sources), see [[#Fra74|Frankel, 1974]], p.{{nnbsp}}234n. The full text (with the misprint) is "Mémoire sur les mouvements de la lumière dans les milieux diaphanes", ''Mémoires de l'Académie des Sciences'', 1st&nbsp;Series, vol.{{nnbsp}}{{serif|X}} (1810), reprinted in ''Oeuvres complètes de Laplace'', vol.{{nnbsp}}12 (Paris, Gauthier-Villars et fils, 1898), [https://archive.org/stream/oeuvrescomplte12lapluoft/#page/267 pp.{{nnbsp}}267–298]. An intermediate version, including the proof but not the appended "Note", appeared as "Sur le mouvement de la lumière dans les milieux diaphanes", ''Mémoires de Physique et de Chimie de la Société d'Arcueil'', vol.{{nnbsp}}2 (1809), [https://books.google.com/books?id=hnJKAAAAYAAJ&pg=PA111 pp.{{nnbsp}}111–142] & [https://books.google.com/books?id=hnJKAAAAYAAJ&pg=PA495 Plate&nbsp;1] (after p.{{nnbsp}}494).</ref>

Young's claim was more general than Laplace's, and likewise upheld Fermat's principle even in the case of extraordinary refraction, in which the rays are generally ''not&nbsp;perpendicular'' to the wavefronts. Unfortunately, however, the omitted middle sentence of the quoted paragraph by Young began "The motion of every undulation must necessarily be in a direction ''perpendicular'' to its surface&nbsp;..." (emphasis added), and was therefore bound to sow confusion rather than clarity.

[[File:Augustin Fresnel.jpg|thumb|Augustin-Jean Fresnel {{nowrap|(1788–1827)}}]]

No such confusion subsists in [[Augustin-Jean Fresnel]]'s "Second Memoir" on double refraction ([[#SecMem|Fresnel,&nbsp;1827]]), which addresses Fermat's principle in several places (without naming Fermat), proceeding from the special case in which rays are normal to wavefronts, to the general case in which rays are paths of least time or stationary time. (In the following summary, page numbers refer to [[#SecMem|Alfred W.{{tsp}}Hobson's translation]].)
* For refraction of a plane wave at parallel incidence on one face of an anisotropic crystalline wedge (pp.{{nnbsp}}291–2), in order to find the "first ray arrived" at an observation point beyond the other face of the wedge, it suffices to treat the rays outside the crystal as normal to the wavefronts, and within the crystal to consider only the parallel wavefronts (whatever the ray direction). So in this case, Fresnel does not attempt to trace the complete ray path.<ref group=Note>In the translation, some lines and symbols are missing from the diagram; the corrected diagram may be found in Fresnel's ''Oeuvres Complètes'', vol.{{nnbsp}}2, [https://books.google.com/books?id=g6tzUG7JmoQC&pg=PA547 p.{{nnbsp}}547].</ref>
* Next, Fresnel considers a ray refracted from a point-source ''M'' inside a crystal, through a point ''A'' on the surface, to an observation point ''B'' outside (pp.{{nnbsp}}294–6). The surface passing through ''B'' and given by the "locus of the disturbances which arrive first" is, according to Huygens' construction, normal to "the ray ''AB'' of swiftest arrival". But this construction requires knowledge of the "surface of the wave" (that is, the secondary wavefront) within the crystal.
* Then he considers a plane wavefront propagating in a medium with non-spherical secondary wavefronts, oriented so that the ray path given by Huygens' construction &ndash; from the source of the secondary wavefront to its point of tangency with the subsequent primary wavefront &ndash; is ''not'' normal to the primary wavefronts (p.{{nnbsp}}296). He shows that this path is nevertheless "the path of quickest arrival of the disturbance" from the earlier primary wavefront to the point of tangency.
* In a later heading (p.{{nnbsp}}305) he declares that "The construction of Huygens, which determines the path of swiftest arrival" is applicable to secondary wavefronts of any shape. He then notes that when we apply Huygens' construction to refraction into a crystal with a two-sheeted secondary wavefront, and draw the lines from the two points of tangency to the center of the secondary wavefront, "we shall have the directions of the two paths of swiftest arrival, and consequently of the ordinary and of the extraordinary ray."
* Under the heading "Definition of the word ''Ray''" (p.{{nnbsp}}309), he concludes that this term must be applied to the line which joins the center of the secondary wave to a point on its surface, whatever the inclination of this line to the surface.
* As a "new consideration" (pp.{{nnbsp}}310–11), he notes that if a plane wavefront is passed through a small hole centered on point ''E'', then the direction ''ED'' of maximum intensity of the resulting beam will be that in which the secondary wave starting from ''E'' will "arrive there the first", and the secondary wavefronts from opposite sides of the hole (equidistant from ''E'') will "arrive at ''D'' in the same time" as each other. This direction is ''not'' assumed to be normal to any wavefront.
Thus Fresnel showed, even for anisotropic media, that the ray path given by Huygens' construction is the path of least time between successive positions of a plane or diverging wavefront, that the ray velocities are the radii of the secondary "wave surface" after unit time, and that a stationary traversal time accounts for the direction of maximum intensity of a beam. However, establishing the general equivalence between Huygens' construction and Fermat's principle would have required further consideration of Fermat's principle in point-to-point terms.

[[Hendrik Lorentz]], in a paper written in 1886 and republished in 1907,{{r|lorentz-1907}} deduced the principle of least time in point-to-point form from Huygens' construction. But the essence of his argument was somewhat obscured by an apparent dependence on [[Luminiferous aether|aether]] and [[Aether drag hypothesis|aether drag]].

Lorentz's work was cited in 1959 by Adriaan J.&nbsp;de Witte, who then offered his own argument, which "although in essence the same, is believed to be more cogent and more general". De&nbsp;Witte's treatment is more original than that description might suggest, although limited to two dimensions; it uses calculus of variations to show that Huygens' construction and Fermat's principle lead to the same [[differential equation]] for the ray path, and that in the case of Fermat's principle, the converse holds. De&nbsp;Witte also noted that "The matter seems to have escaped treatment in textbooks."<ref>[[#deWitte|De Witte, 1959]], esp. pp.{{nnbsp}}293n,{{nnbsp}}298.</ref>