Lenses and Waves

Kepler began with the understanding
of the nature of light as the surface of an
expanding sphere laid down in the
opening chapter of Paralipomena (Figure
37). ABMK is the section of a physical ray
obliquely incident on the surface BC and
refracted towards QBMR. According to
Kepler the angle of deviation must be
proportional to the angle of incidence.
THE 'PROJET' OF 1672 121
Figure 37 The final stage of Kepler’s
analysis of refraction
This condition is met when only the (surface)density of the refracting
medium is assumed to be effective. With increasing obliquity, BM increases
and therefore the resistance met by the light increases. Now “… there is
more density in BM than in LM ...” so that the proportion LM to BM must be
added as a factor of refraction onto the proportionality of angles of
incidence and deviation. 59 However, the proportion LM to BM – or sec i –
implied a paradox. Horizontal rays would be refracted at an infinitely large
angle. Kepler therefore changed his perspective and now considered BR of
the refracted ray. He concluded that the secans of the angle at the upper
surface of the denser medium ‘plays a part’ in refraction. 60 Refraction was
thus a composite of two factors: the proportionality of i-r to i and the
proportionality of i-r to sec r – in other words: i-r = c·i·sec r, where c is some
constant.
In proposition 8, Kepler gave instructions how to apply this analysis to
calculate angles of refraction. It is in the form of a ‘problem’, a procedural
statement of the sort the later Dioptrice was composed of, as we saw above in
section 2.2.1. After all, Kepler’s struggle had not yielded a general ‘measure
of refraction’ independent of specific media and transcending measurements.
First, both factors are determined for the medium by means of one known
pair of incident and refracted rays. Then the angle of refraction for any other
angle of incidence is computed. By means of an example, Kepler calculated a
table for refraction from air into water. The values differed somewhat from
Witelo’s data which Kepler had plied so rigorously in the previous sections.
This time he was more tolerant: “This tiny discrepancy should not move you;
believe me: below such a degree of precision, experience does not go in this
not very well-fitted business.” 61 Moreover, he (correctly) suspected that
Witelo had modified his table on the basis of Ptolemy’s false supposition
that the secondary differences of the angles are constant. “Therefore, the
fault lies in Witelo’s refractions”, and Kepler proceeded to use his own result
to consider atmospheric refraction. 62 Although the empirical correctness was
59 Kepler, Paralipomena, 111 (KGW2, 105). “Plùs igitur densitatis est in BM, quàm in LM.”
60 Kepler, Paralipomena, 113 (KGW2, 107). “..., sciendum igitur, eorum angulorum incidentiae secantes concurrere ad
mensuram refractionum, qui constituuntur ad superficiem in medio densiori.”
61 Kepler, Paralipomena, 116 (KGW2, 109). “Neque te moveat tantilla discrepantia, credas mihi, infra tantam
subtilitatem, experientiam in hac minus apt materia non descendere.”
62 Kepler, Paralipomena, 116 (KGW2, 109). “Ergò in Vitellionis refractionibus culpa haeret.”