Lenses and Waves

256 CHAPTER 7
Huygens was the first (and for a long time the only one) to pursue the
question raised by Kepler right after the invention of the telescope: how can
we understand its working in a mathematical way? Students of dioptrics like
Descartes, Barrow, and Newton focused on solving sophisticated
mathematical problems like determining aplanatic surfaces and analyzing
optical imagery. They did not elaborate their findings to explain the
dioptrical properties of ordinary lenses and their configurations.
Astronomical observers, starting with Galileo, did not elaborate a dioptrical
theory of telescopes either. Only when the telescope was turned, towards
1670, into an instrument of precision did its users like Flamsteed and Picard
begin to bother about questions of dioptrics. Without Huygens’
mathematical proficiency they could not, however, obtain the rigor and
generality of Tractatus and De Aberratione. But Huygens was not of help, he
never came to publish his dioptrics.
Despite the fact that he always had an open eye for practical implications,
the orientation on instruments characteristic of Dioptrica cannot be directly
generalized. The organ did not direct his studies of consonance, as clocks did
not direct his studies of motion. 1 Still, in a broader sense Dioptrica does reveal
a particular feature of Huygens’ science. Whereas Descartes contented
himself with establishing the principles of refraction and perfect vision,
Huygens applied the sine law to establish the properties of actual lenses and
telescopes. Whereas others analyzed the mathematics of lenses in order to
find perfectly focusing surfaces, he did so in order to understand the
properties of real lenses and fathom their imperfections mathematically.
What Huygens did in Dioptrica – apart from the practical relevance of his
pursuits – was elaborating mathematical theory by applying general principles
to specific problems of real objects.
Huygens ‘applied geometry to matter’, to use his phrase in Traité de la
Lumière. Real, rather than ideal matter. Application in the sense of elaborating
established mathematical theory for particular cases, rather than
mathematization of new phenomena in the sense Newton did with colors.
Even in his theories of impact and circular motion he substantially built
upon mathematical foundations already laid by Galileo. As contrasted to
Newton, he mathematized no phenomena that had not already latently been
mathematized. Rather than establishing an investigation into the physics of
consonance, he elaborated the mathematics of the coincidence theory. In
dioptrics, he confined himself to the analysis of the properties of refracted
rays and left colors for what they were. Brilliantly pursuing mathematical
reasoning, he rarely went beyond the established boundaries of the
mathematical sciences.
One cannot escape the impression that the elaboration of mathematical
theory for particular problems interested him more than laying new
1
Although these have never, to my knowledge, been studied from the viewpoint of the relationship
between theory and practice.