Editing Fermat's principle (section)

=== 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.