Lenses and Waves

202 CHAPTER 5
With the full elaboration of his theory in Traité de la Lumière, Huygens
showed that with his principle propagated waves could be constructed in any
situation. Only the speed of propagation, as it depended upon the medium
traversed, needed to be varied. In this way he derived all observable
properties of light rays from one and the same principle in a mutually
consistent way. This reduction was what he understood by explanation.
Reducing the properties of light rays to Huygens’ principle was explaining
these properties mechanistically, because the principle explicated the
essentials of successive impact in ethereal particles. The validity of this
‘principal foundation’ rested upon the fact that the laws of optics could be
reduced to it. In other words, it did not rest upon the appropriateness of
‘raisons de mechanique’, but on the plausibility of mathematical inference.
In Huygens’ wave theory three levels can be distinguished: a mechanistic
model of colliding particles, the laws of optics and – in between – Huygens’
principle. As the mathematical representation of the mechanistic nature of
wave propagation, Huygens’ principle serves as a intermediary of a special
kind between the nature of light and the laws of optics. It was the
indispensable link between Huygens’ mechanistic picture of collisions of
ethereal particles and the mathematical laws of light rays. In the light of
seventeenth-century geometrical optics, where the laws of optics functioned
as the postulates or principles of mathematical science, Huygens’ principle of
wave propagation can be called a law of optics. Not in the modern sense of a
law of nature in physical science, but in a then traditional sense. Remember
that the sine law and the like were rarely called laws then, but rules, measures
or properties. In the mathematical science of optics Huygens had disclosed a
new law, a more fundamental one to which the various properties of light
propagation were subordinated to. However, this new ‘law’ was of an entirely
different nature than the traditional principles of optics. Huygens’ principle
did not describe the behavior of rays but the behavior of waves; it was a
mathematical law describing the behavior of unobservable entities.
Comprehended in this way, Huygens’ principle was a novel element in the
mathematical science of optics. Huygens’ principle not only unified ordinary
and strange refraction, it unified all properties of light rays. It was a more
general law and a law of different character at the same time, describing the
behavior of unobservable waves mathematically.
One might say that Huygens had brought geometrical optics to a new
level, that of microphysics. He focused on the geometrical constructions
with his principle and did not spell out its mechanistic underpinning. In
Traité de la Lumière, waves have taken the place of rays. Waves are entities
with well-defined mathematical properties, the causes of which are explained
rather informally, like in Barrow’s elucidations. Huygens switched to the
mathematical consideration of waves in a matter-of-course way. He applied
geometry to these unobservable entities with the same ease as he applied it to
observable balls and pendulums. In his wave theory he extended Galileo’s
mathematical physics of observables to that of unobservables. As one