Editing Fermat's principle (section)

=== 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.