Lenses and Waves

76 CHAPTER 3
Huygens supposed that the circle of aberration XV of a lens system is
mainly produced by the objective lens AB (Figure 31). The ocular lens PO
barely increases the diameter of the circle and could therefore be considered
to have a perfect focus. He considered the opening BC of the objective lens
required to maintain a constant circle of aberration when the focal distance
CD of this lens is changed. He proved that the proportion CD 3 : CB 4 should
remain constant. 100 Finally, the quality of images will be maintained upon
changing the ocular lens, when the proportion OD : 4 CD between the focal
distances of both lenses is maintained. 101 Again, Huygens converted these
proportions into a table of numerical values, listing the optimal values of the
focal distances of both lenses and the opening of the objective, as well as the
resulting magnification of the system. 102 This table concluded De Aberratione.
Huygens’ theoretical accomplishments in De Aberratione are beyond dispute.
Like the theory of focal distances and magnification of Tractatus, his theory
of spherical aberration was rigorous and general. And again his theoretical
studies were aimed at understanding the telescope; in this case, at
understanding how a system of lenses produces spherical aberration.
Huygens could claim that he understood why an opening of such-and-such
dimensions maintained the quality of images.
His results were couched in two tables listing the required components to
make these optimal systems, in a way quite comparable to the ones found in
Bolantio’s manuscript. They prescribed how to assemble a telescope without
presupposing theoretical knowledge of dioptrics. The difference is that
Huygens’ tables were derived from his mathematical theory of lenses instead
of a record of experiential knowledge. The table prescribing the aperture of
telescopes was not gained by some implicit rule of thumb, but was based on
an explicit theorem derived from dioptrical properties of lenses. Huygens
could prove that the openings he prescribed were optimal. Whether this
worked in practice remains to be seen. At least he could claim that he could
calculate beforehand how to adjust the components of a telescope when its
length was changed, thus avoiding a renewed process of trial-and-error.
Huygens had realized the goal of De Aberratione. He had demonstrated
that the aberrations of spherical lenses could be made to cancel out.
100 OC13, 342-345.
101 OC13, 348-351.
102 OC13, 350-353.
Figure 31 Aberration produced by a Keplerian configuration.