Lenses and Waves

120 CHAPTER 4
experience. 53 He tried some more ideas flowing
from this geometry, including some ways of
evaluating the ‘refractaria’ - the locus of images
where the points of a line are percieved, D for
point L, I for point M, etcetera. 54 All ideas were
refuted by experience and Kepler abandoned his
attempt of finding a measure of refraction on the
basis of an analysis of the physics of light rays.
In the next three sections, Kepler temporarily
ignored the causes of refraction and focused on
finding mathematical regularities in the given
angles of incidence and refraction. Building on the
known properties of reflection, he tried certain
analogies between reflection and refraction. Kepler
argued that in the case of refraction in a medium
with infinite density, all rays must be refracted into
the perpendicular. He then correlated this case to
reflection by a parabolic mirror with rays coming
from its focus. This led him to consider the
relationship of conic sections with refraction. He
constructed a diagram of angles of incidence and
Figure 36 The first stage of
Kepler’s attack of
refraction.
refraction and considered the intersection of the accompanying rays. The
resulting curve is similar to a hyperbola, but points from where the rays
come are not the matching foci, so Kepler dismissed this attempt as well.
This and other trials with conic sections – including the effort to construct
an anaclastic curve – still did not give Kepler a correct ‘measure of
refractions’ and he abandoned this line of thought as well.
Finally, Kepler returned to his causal analysis of refraction of chapter 1 to
query whether - “may God look kindly upon us” - this would yield the
measure of refraction. 55 As contrasted to the ray analysis of the first stage, he
now considered the interaction of the surface of light with the surface of the
refracting medium. 56 Kepler warned beforehand he would perhaps stray
somewhat from his goal of finding the measure of refractions in its causes,
and halfway through his exercise he would acknowledge “In demonstrating
the true cause of this directly and a priori, I am stuck.” 57 He did not formally
deduce a ‘measure’ from the causes of refraction, but rather had employed
(in Buchdahl’s words) “physical considerations to guide the intuitive search
for responsible factors relevant to the result.” 58
53
Kepler, Paralipomena, 86 (KGW2, 86). “Hic modus refutatur experientiâ: ...”.
54
Kepler, Paralipomena, 88-89 (KGW2, 87-88).
55
Kepler, Paralipomena, 110 (KGW2, 104). “quod Deus benè vertat”
56
Kepler, Paralipomena, 110-114 (KGW2, 104-108).
57
Kepler, Paralipomena, 110 and 113 (KGW2, 104 and 107). “Etsi enim à scopo forsan etiamnum nonnihil
aberrabimus: ...” and “In genuina huius rei causa directè et à priori demonstranda haereo.”
58
Buchdahl, “Methodological aspects”, 291.