Lenses and Waves

1653 - TRACTATUS 33
focused on problems pertaining to their configurations in actual telescopes.
Like Kepler, Huygens intended to found the dioptrical properties on a sound
mathematical basis. Whether a continuation of Dioptrice was his actual goal,
can only be surmised as he did not explicitly refer to it in such a
programmatic sense. Huygens did know Dioptrice, it had been on the reading
list of his mathematics tutor Stampioen and much later he commended it to
his brother Constantijn as the best introduction to dioptrics. 80 Huygens did
not have much to offer that was not already known. Tractatus covered more
types of lenses but the eventual results regarding the focal distances of lenses
and the magnifying properties did not differ much from Dioptrice. The crucial
difference is that Huygens founded his results on a general and exact theory
of focal distances. It rigorously proved Kepler’s results. He had the exact law
of refraction at his disposal and thus could be exact where Kepler necessarily
had to leave his readers with approximate answers.
Perspectiva and the telescope
At the same time when Kepler wrote Dioptrice, two other scholars devised an
account of the telescope. Della Porta’s ‘De telescopio’ remained
unpublished, De Domini’s De Radiis Visus et Lucis was published in 1611.
Both were based on perspectivist theory of image formation. Before I go on
to discuss the impact of the sine law on dioptrics, I briefly discuss these in
order to make it clear why that perspectivist theory was intrinsically
inadequate to account fully for the effect of lenses.
Shortly before his death, Della Porta extended his theory of lenses of De
Refractione to telescopes in a manuscript ‘De telescopio’. 81 It reveals the
problems lenses posed for perspectivist theory of image formation. In order
to determine the place where an object is seen, perspectiva used the cathetus
rule. The cathetus is the line through the object point, perpendicular to the
reflecting or refracting surface. The cathetus rule states that the image is the
intersection of the ray entering the eye and the cathetus. Modern Keplerian
theory shows that, although valid in many cases, this rule turns out to break
down for curved surfaces in particular. To account for images of lenses
another problem turns up. As a lens refracts a ray twice, this seems to imply
that the rule has to be applied twice also. Della Porta avoided this problem
by considering only one cathetus.
Della Porta considered a lens in terms of refracting spheres, as he had
done in De Refractione (Figure 17). The dotted lines indicate such spheres and
the lens dcgf is formed by their overlap. 82 The object ab is perceived as
follows: a ray from point a is refracted along cd to the eye. Della Porta drew
the cathetus ka of the lower surface of the lens, which also is its radius.
When produced, the ray entering the eye intersects the cathetus in point h,
80
OC1, 6 (Stampioen’s list of recommended readings spans pages 5-10) and OC6, 215.
81
Della Porta’s account of refraction by spheres and lenses in De refractione is discussed in Lindberg,
“Optics in 16th century Italy”, 143-146.
82
Della Porta, De Telescopio, 113-114.