Lenses and Waves

1655-1672 - DE ABERRATIONE 101
circular motion. He often couched his thoughts on circular motion in some
mechanical form. And he designed several clocks that embodied his
theoretical findings. As regards his original pendulum clock he reaped the
rewards of his study by equipping it with cheeks that gave its bob an
isochronous path.
If instruments did not guide Huygens’ other studies the way they did in
dioptrics, his approach to them was nevertheless similar. Horologium
Oscillatorium of 1673 does not just describe the pendulum clock and the ideal
cycloidal path, but also gives the mathematical theory of motion embodied in
it. Going beyond the mere necessities of explaining its mechanical working –
as in Dioptrica – he elaborated his theories of circular motion, evolutes and
physical pendulums. Of the achievements of 1659, Horologium Oscillatorium
included the study of curvilinear fall and cycloidal motion, transformed into
a direct and refined derivation, but it listed only the resulting propositions of
his study of circular motion and the conical clock. In addition, it contained a
discussion of physical pendulums. Huygens imaginatively applied the insight
that a system of bodies can be considered as a single body concentrated in
the center of gravity, to a physical pendulum considered to be resolved into
its constituent parts independently. With this he could express the motion of
the pendulum by means of the accelerated motion of its parts. Next he
compared the physical pendulum to an isochronous simple pendulum,
deriving an expression for the length of the latter in terms of the length and
the weights of the parts of the former. 210
His organ likewise rested on an sound and even elegant theory of
consonance. In this way he showed the solid theoretical basis on which his
inventions rested, showing at the same time that he was not a mere
empiricist but a learned inventor. 211 De Aberratione stands out among
Huygens’ studies in that he developed theory with the explicit aim of
improving an instrument. Earlier, he had proven the working of his eyepiece
on a mathematical basis, but he had not been able to demonstrate that it was
the best configuration possible. In De Aberratione Huygens set out to design a
configuration of lenses that he could prove mathematically was the best one
possible.
Huygens was not unique for trying to solve a practical problem by means
of theory. Descartes’ a-spherical lenses were meant to serve as a solution to
the same problem Huygens attacked. Descartes had tried to realize his design
by thinking up a device fit for making those lenses. Examples from other
fields can be found without much effort; the problem of finding longitude at
sea is only the first to come to mind. The seventeenth century is pervaded by
scholars who believed theory could or should be of practical use. The special
thing about De Aberratione is the way Huygens set out to solve the problem
of spherical aberration. His starting point consisted of the mathematical
210 Westfall, Force, 165-167.
211 Cohen, Quantifying music, 224.