Lenses and Waves

26 CHAPTER 2
reply, Kepler briefly explained how lenses refract rays of light so that they
can produced magnified images. 53 The explanation in Dissertatio was only a
sketch, but the message was clear. The telescope was a remarkable invention,
but its working needed mathematical clarification. A theory of the telescope
was called for. Within a few months, Kepler developed one. In September
1610, he finished the manuscript of Dioptrice, published the next year.
“Some have disputed over the priority of its invention, others rather applied themselves
to the perfection of the instrument, as there chance mainly counted, here reason
dominated. But Galileo scored the greatest triumph by exploring its use to disclose
secrets, because zeal procured him with the design and fortune has not withheld him
the success. I, driven by an honest emulation, have shown the mathematicians a new
field to expose their acuteness, in which the causes and principles are retraced to the
laws of geometry, the effects of which are so awaited with much impatience and are of
such pleasing diversity.” 54
The goal of Dioptrice was to provide a mathematical account of the working
of the telescope. In Kepler’s view, the working of any instrument used in
astronomy should be understood precisely. A decade earlier, he had
approached the puzzling properties of the pinhole images used in the
observation of solar eclipses. His answer had been a new theory of image
formation, which he had published in Paralipomena (1604). In Dioptrice, Kepler
applied this theory to lenses in order to determine the dioptrical properties
of the telescope. Dioptrice had one substantial shortcoming: Kepler knew that
he did not know the exact law of refraction. He used an approximate rule
instead.
2.2.1 KEPLER AND THE MATHEMATICS OF LENSES
Kepler’s concerns in Paralipomena were induced by a problem of astronomical
observation. In 1598, Tycho Brahe had reported an anomalous observation
of the apparent size of the moon during a solar eclipse. 55 Brahe used a
pinhole to project the image of the eclipsed sun. When he measured the
diameter of the projection he realized that “the moon during a solar eclipse
does not appear to be the same size as it appears at other times during full
moons when it is equally far away”. 56 He tried to produce consistent values
by applying some ad hoc corrections to his measurements. 57 Kepler took a
different approach, analyzing mathematically the way the image was
produced. He had known the anomaly of pinhole images for some time
53
Kepler, Conversation, [19-21].
54
Kepler, Dioptrice, dedication (KGW4, 331). “… circaque eam alij de palma primae inventionis certarent,
alij de perfectione instrumenti sese jactarent amplius, quod ibi casus potissimum insit, hic Ratio
dominetur: GALILAEUS vero super usu patefacto in perquirendis arcanis Astronomicis speciosissimum
triumphum ageret; ut cui consilium suppeditaverat industria, nec successum negaverat fortuna: Ego
doctus honesta quadam aemulatione novum Mathematicis campum aperui exerendi vim ingenij, hoc est
causarum lege geometrica demonstrandarum, quibus tam exoptati, tam jucunda varietate multiplices
effectus inniterentur.”
55
Straker, “Kepler’s theory of pinhole images”, 276-278.
56
Cited and translated in: Straker, “Kepler’s theory of pinhole images”, 278.
57
Straker, “Kepler’s theory of pinhole images”, 275-276; 280-282.