Lenses and Waves

166 CHAPTER 5
a certain amount of time, that is, as optical paths. He then applied this to
caustics. In this case he considered a number of rays. He measured out equal
times covered along different rays and recognized the relationship between
caustics and involutes, curves he was fully acquainted with. Somewhere along
the line, Huygens realized that in constructing waves this way he was
assuming that rays are always normal to waves. If that is so, the notion of
wavelets spreading in all directions from the points of incidence and
subsequently forming a propagated wave by their enveloping curve must
have come up as a justification. 22
Huygens’ account of caustics resolved two ambiguities in Pardies’
derivation of refraction. Pardies had determined the direction into the
refracted wave propagates by constructing the refracted ray (Figure 56 on
page 153). The line sections Ce, ee, etc. on the refracted rays are regarded as
the intervals by which the wave proceeds in a given time. The resulting wave
is not spherical anymore, but this is not accounted for any further. The
meaning of the curve in terms of light waves thus remains vague.
Furthermore, as Shapiro has pointed out: “Pardies has not explained why at
the point of refraction the wave should be refracted in one direction, … He has
simply assumed that refraction occurs. Therefore, he has demonstrated only
that if the wave is refracted, then it must be propagated in the new medium
in a direction such that the rays are always normal to the wave fronts.” 23
Huygens now stated that light, in the form of ‘particular waves’, spreads in all
directions from the points of refraction. The propagated wave can be
constructed by drawing the envelope of the wavelets, even if the wave is not
spherical. Implicitly, Huygens stated that the wave is a surface of constant
phase, i.e. the locus of points where wavelets unite. It remains to be seen
whether Huygens was explicitly reconsidering Pardies’ theory of light at this
moment. It is unclear, for example, whether the ‘surprising’ observation that
‘rays not tending to one center, can always cut the waves at right angles’ had
given rise to the study of ovals and caustics. 24
Irrespective of the question of whether he was explicitly tackling
problems with Pardies’ theory, Huygens had begun to focus on distances
covered by light in a specific time. And irrespective of the question of
whether, and if so how, his principle of wave propagation arose from the
ensuing analysis of caustics, he realized that both waves and rays are
22 In other words, I tend to disagree with Shapiro’s view that this ‘most subtle refinement of Huygens’
optics’ cannot have been the starting point for the formulation of Huygens’ principle. (241) I do not
consider the equality of optical paths to have been derived from his theory of light (232), but rather to be
Huygens’ starting point in these studies, which he subsequently, and rather implicitly, justified by a vague
notion of his principle. I suspect that Shapiro has been misled by following the text of Traité de la Lumière,
that is, the analysis of the enveloping wave refracted by a spherical surface, which is indeed the most
subtle refinement and application of Huygens’ theory of light. It must, however, be of a much later date
as this case is not found in the 1667 notes. Shapiro, “Kinematic optics”, 231-236.
23 Shapiro, “Kinematic optics”, 215-217. Emphasis in the original.
24 Ziggelaar states, without argument, that caustics posed a crucial objection to Pardies’ theory and that
this induced Huygens to formulate his own theory: Ziggelaar, “How”, 186-187.