Lenses and Waves

1677-1679 –WAVES OF LIGHT 165
On the next two pages, the issue at stake does become clear. 19 Here he
considered the same wave, but now propagating from air to the glass
between E and D. The incident rays intersect in C (Figure 61). When the
whole wave has passed the refracting surface, the wave XxxxxE is formed in
the glass. The accompanying text reads:
“The common tangent curve of all the particular waves will be the propagation of the
principal wave in glass. Therefore, the straight lines which cut this tangent curve at right
angles will be the refracted rays. These, however, are given otherwise. Therefore these
will cut that curve at right angles. Therefore the curve is the involute of the other curve
which is the common tangent of these rays. It is sufficient to know that the waves are
propagated along these straight lines. But since the lines must cut the waves at right
angles, it can appear surprising how the lines, not tending to one center, can always cut
the waves at right angles. But this is now explained by the involute.” 20
This text makes two things clear. In the first place, Huygens finally says what
problem had been involved in the preceding exercises. In both cases
discussed, rays do not intersect in one point after refraction. The question
therefore is how the accompanying wave ought to be imagined. Apparently,
as we shall soon see, caustics (or aberration in general) also raised questions
with Pardies’ theory, as it is not immediately clear whether or how rays are
normal to waves. Huygens had settled the matter by means of involutes. The
refracted rays form a curve AB, like the curve VHN in Figure 60. The wave
XxxxxE is the involute of this curve.
In the second place, Huygens was applying a new conception of wave
propagation. The particular waves he talks about are not drawn, but are
thought to be the various spherical waves spreading in all directions through
the glass around the points of incidence. At the time the wave in air reaches
E, these wavelets have covered the distance to points x. Their tangent is
XxxxxE, the propagation of the principal wave. Huygens leaves out this step
and immediately goes on to draw the ‘straight lines’ along which it is
propagated, the rays that is. He can do so because these are ‘given otherwise’,
namely by the sine law. So, instead of determining the refracted rays by
constructing the propagated wave, he determines the propagated wave by
constructing the refracted rays. As I see it, instead of applying his principle to
construct the propagated ray, Huygens was using it to justify the
construction by means of refracted rays.
The insight underlying this justification does not go beyond a small
sketch one page further down (Figure 58 on page 162). 21 We cannot be
certain to what extent the insight was already in his mind. I believe it was
beginning to take shape when he was analyzing caustics in figures 59 to 61. I
find the preceding study of Fermat’s principle and of aplanatic surfaces
telling. Huygens was beginning to consider rays in terms of a path covered in
19 Hug9, 41v and 42r. OC19, 421 §3 and 422 respectively. I only discuss 42r.
20 OC19, 422; translation from Shapiro, “Kinematic optics”, 236.
21 Hug9, 43r. On the intermediate page 42v he applies it to the propagation of a wave crossing an aplanatic
surface (OC19, 425-426 §2), but this apparently is much later as Huygens dated it 24 March 1678.