Lenses and Waves

efracted wave and thus
perpendicular to its direction
of propagation Cee. Thus
Cee is the refracted ray for
incident ray ccC. It is easily
shown that the sine law
holds.
Now compare Descartes’
derivation and Huygens’
extension of it to strange
refraction. Descartes
assumed that the parallel
component of the ray was
conserved. He did not say
anything about the
perpendicular component.
This accords with Huygens’
THE 'PROJET' OF 1672 153
construction, which adds a ‘lateral’ component to an ordinarily refracted ray.
The second assumption of Descartes’ derivation was a constant proportion
of the motions of the ray before and after refraction. Pardies also assumed
such a constant proportion, but exactly the other way around. Waves move
faster in air than in glass, whereas in Descartes’ derivation rays necessarily
move fastest in glass. Consequently, a Cartesian derivation contradicts a
Pardies-like explanation of refraction. Moreover, in Pardies’ derivation of the
sine law, both components of the ray have changed after refraction,
rendering the Cartesian analysis meaningless. 151
How could so gifted a man as Huygens
overlook such an obvious inconsistency?
We should bear in mind that, in Dioptrica,
Huygens never used the Cartesian circle
diagram (Figure 40 on page 127). He always
visualized the sine law as a ‘cathetus’
construction, where CG and CD are
constructed according to the ratio of sines
(Figure 57). A similar approach is also
suggested by DS and DF of the ordinarily
refracted ray (Figure 54 on page 149). The
details of Descartes’ derivation need not
Figure 56 Ango’s explanation of refraction.
Figure 57 The sine law in Tractatus.
have been on top of his head when Huygens added his strange component.
Although he stayed closer to the drift of Descartes’ derivation as compared
to Bartholinus – who merely extended the circle diagram – he nevertheless
151 In Ango’s diagram the proportion of speeds is directly represented by the ratio Kn : Km and
subsequently by the distances cc and ee traversed by the waves of light. These distances can, in its turn, be
analyzed in parallel and perpendicular components, but both change according to the figure. The
assumption visini = vrsinr is meaningless.