Lenses and Waves

1653 - TRACTATUS 31
incidence; the angle between the incident ray, produced beyond the
refracting surface, and the refracted ray.
Kepler began with a discussion
of the focal distances of planoconvex
lenses (Figure 15). A ray
HG is incident on a convex surface
with radius AC, the angle of
incidence is GAC. As the angle of
deviation is one third of this, HG
will be refracted towards F, with AC : AF = 1 : 2. 71 The focal distance is
therefore approximately three times the radius of the convex face.
Analogously, he argued that the focal distance of a plano-convex lens, the
plane face turned towards the incident rays, is approximately twice the radius
of curvature. For other cases Kepler established only rough estimations. If
convergent rays are incident on the plane side of a plano-convex lens, the
refracted rays intersect the axis within the focal distance. Combining these
three theorems, Kepler showed that the focal distance of a bi-convex lens is
both smaller than three times the radius of the anterior side and twice the
radius of the posterior side. In the case of an equi-convex lens, this comes
down to a focal distance approximately equal to the radius of its sides. 72
Kepler did not determine the focal distance of a concave lens, he only
showed that rays diverge after refraction. 73
On this basis, the properties of images formed by lenses are easily found.
The image DBF of an extended object CAE through a bi-convex lens GH is
formed at focal distance (Figure 16). The picture is inversed as the rays from
C are refracted towards D, etcetera. As the focal distance is roughly the radius
of any side, the magnitudes of object and image will be in a proportion equal
to their respective distances to the lens. 74 In Dioptrice, Kepler briefly reiterated
his theory of vision. On the one hand, so he said in the dedication, he did so
for the sake of completeness, on the other hand because some readers had
trouble understanding his account in Paralipomena. 75 He explained that a
perfectly focusing surface was not spherical, but should be hyperbolic, like
the crystalline humor of the eye was. 76 On the basis of his theory of the
retinal image, he explained the effect of a lens placed before the eye once
more. Depending upon the position of the eye with respect to the focal
distance, the object will be perceived sharply. 77 When the eye is placed not
too far from the focus, a magnified image will be perceived.
71 Kepler, Dioptrice, 11 (KGW4, 363).
72 Kepler, Dioptrice, 12-15 (KGW4, 363-367).
73 Kepler, Dioptrice, 45-49 (KGW4, 388-393).
74 Kepler, Dioptrice, 16-18 (KGW4, 367-369).
75 Kepler, Dioptrice, dedication (KGW4, 335).
76 Kepler, Dioptrice, 21-24 (KGW4, 371-372).
77 Kepler, Dioptrice, 35-42 (KGW4, 381-387).
Figure 15 Focal distance of a plano-convex lens