Lenses and Waves

162 CHAPTER 5
Cartesian ovals and caustics also conducted on these pages. In my view, the
solution to the problem of strange refraction came directly out of this line of
thought, although it may be argued that at some point some break occurred. 9
The editors failed to reproduce the material on the five pages preceding the
EUPHKA. It is indisputably related to the solution and, although somewhat
obscure, it clarifies the way the solution may have taken shape in Huygens’
mind.
Both Huygens’ account of caustics and his
explanation of strange refraction depend upon
Huygens new conception of wave propagation.
Yet, it remains implicit throughout the notes
involved. Only one or two tiny sketches reveal
that an adjustment of Pardies’ wave theory had
taken shape in his mind. The crucial insight is
first found in the sketch reproduced in Figure
58. All points on a wave are centers of a
multitude of wavelets spreading in all directions;
the tangent to these wavelets is the propagated
wave. Only in Traité de la Lumière did Huygens
elaborate his principle of wave propagation and
its application to the behavior of light rays.
Figure 58 Huygens’ principle.
To my knowledge, only two previous historians have consulted parts of
the manuscripts reproduced in volume 19 of the Oeuvres Complètes. They were
puzzled. According to Shapiro, they contain “the most subtle refinement of
Huygens’ optics” which cannot have been its starting point. 10 Ziggelaar has
suggested that they reflect the “flash of genius” in which Huygens found his
principle, that he then applied to the sophisticated problem of caustics. 11 In
my view, rather than a matter of application, the principle gradually emerged
from the analysis of caustics. By following the manuscript material, we may
find out what the “flash of genius” was – if it was one at all – and what
sparked it.
5.1.1 A FIRST EUPHKA
The pages begin with a drawing of a ray refracted by a plane surface,
accompanied by a formulation of Fermat’s principle of least time: a ray is
refracted in such a way that light travels between two points in different
media in minimal time. 12 On the next pages Huygens used the ensuing
equation for the lengths of the two parts of the refracted ray to construct a
Cartesian oval, the curve that refracts rays from one point to exactly a
9
Hug9, 42v is clearly written later, as it is dated 24 March 1678. The following three pages contain some
scattered sketches and calculations.
10
Shapiro, “Kinematic optics”, 241.
11
Ziggelaar, “How”, 187.
12
Hug9, 39r; §1on OC19, 416.