Lenses and Waves

176 CHAPTER 5
surface in points AKKKB successively, producing wavelets spreading in all
directions through the refracting medium around these points. When the
whole wave has reached the surface – when C arrives in B – around A a
wavelet will have propagated over the distance AN. The common tangent NB
of all wavelets around the points of incidence is the propagated principal
wave.
The sine law of refraction easily follows. CB represents the speed of light
in the upper medium, but also the sine of the angle BAC, equal to the angle
of incidence DAE. Likewise, AN is the speed in the lower medium and the
sine of angle ABN, equal to the angle of refraction FAN. The assumption that
CB and AN are in constant proportion directly yields the sine law. In the same
way, Huygens could derive the law of reflection by considering only the
propagation of waves. Assuming that the motion of light rebounds at a
reflecting surface, the tangent of wavelets spreading around the points of
reflection is constructed and the equality of the angles of incidence and
reflection readily follows.
Huygens’ theory was simpler and contained less ambiguities than Pardies’.
He had reduced waves to a single property of light, its speed of propagation.
Waves are the effect of an action spreading with a certain speed. In his
derivation of the sine law, Huygens did not have to presume that a wave
refracts. He only had to consider the consequence of an alteration in the
speed of propagation. The curve (or line) resulting from his construction has
an unambiguous meaning, established by his principle of wave propagation.
He preserved the premise that rays are normal to waves, at least in the case
of spherical waves. As a ray is the path traveled by a point of a wave in a
specific time, it is a direct consequence of the fact that the principal wave is
the tangent of secondary waves.
Explaining strange refraction
Precisely by this reduction of waves to speed of propagation, the puzzle of
strange refraction had been solved. In Iceland crystal light propagates with
differing speeds in differing directions. In his notes, Huygens had not
explained why spheroidal waves account for the refracted perpendicular, nor
why light propagates spheroidally in Iceland crystal in the first place. In the
fifth chapter of Traité de la Lumière, Huygens elaborated his discovery of 6
August 1677. He began with a description of the crystal and its peculiar
properties. It displays double refraction, so supposedly light propagates
through the crystal in two different ways. The first one was regular and
produced ordinary refraction in agreement with the sine law. The other one
was irregular, as a perpendicular ray was refracted. To account for this
strange phenomenon, Huygens “wanted to try what elliptical, or better
speaking spheroidal, waves would do”. 49 In other words: try and see what
would happen when light propagates in this direction faster than that.
49 Traité, 58. “Quant à l’autre émanation qui devoit produire la refraction irreguliere, je voulus essaier ce
que feroient des ondes Elliptiques, ou pour mieux dire spheroïdes; …”