Lenses and Waves

38 CHAPTER 2
minute or First Draught of the Optiques” of 1646 (the most complete
elaboration of his optics) included several chapters on lenses and
telescopes. 101 He did not, however, make the most of his knowledge of the
sine law. The account consisted of qualitative theorems – without proof and
often dubious – which applied mostly to single rays refracted by lenses.
Despite the presence of an exact law of refraction, Hobbes’ account (if
published) would have been no match for Dioptrice.
With the exact law of refraction established and published, the road
might seem open for a follow-up of Dioptrice in the form of an exact theory
of the dioptrical properties of spherical lenses. It was not to be, for various
reasons. First of all the sine law became generally known and accepted only
around 1660. 102 This delay may have been caused by a slow distribution of
Descartes’ works – and this maybe partly because La Dioptrique was written
in French – or the bad odor his ideas were in. As late as 1663, in Optica
promota, Gregory showed that the ellipse and hyperbola are aplanatic without
using the sine law. In 1647, Cavalieri extended the theory of Dioptrice to some
more types of lenses, using Kepler’s original rule. As the title Exercitationes
geometricae sex suggests, this was an exercise in mathematics not aimed at
furthering the understanding of the telescope. In this regard, Cavalieri was
not an exception.
Further, and more importantly, mathematicians addressed questions
raised in Kepler’s Paralipomena rather than in his Dioptrice, to wit abstract
optical imagery pertaining to Kepler’s theory of image formation, and the
‘anaclastic’ problem that had been put in a different light by that theory. The
anaclastic problem, or ‘Alhacen’s problem’, is closely related to the
determination of aplanatic surfaces: to find the point of reflection or
refraction of a ray passing from a given point to another. 103 When all rays are
considered, as is relevant in Kepler’s theory of image formation, to find these
points means determining the aplanatic surface. In this theory images are
formed by the focusing of bundles of rays, and in most cases of reflection
and refraction the image of a point source will not be a point. The properties
of these images became an important subject of study in seventeenth-century
geometrical optics. In Optica promota, Gregory extended the theory of
Paralipomena with his contributions to the theory of optical imagery and his
determination of aplanatic surfaces. From this viewpoint, La Dioptrique
embroidered on Paralipomena rather than Dioptrice.
The seventeenth-century study of these topics reached its highpoint in
the lectures Barrow and, later, Newton delivered at the university of
Cambridge. Barrow’s lectures were published in 1669, those of Newton
remained unpublished during his lifetime. With Huygens’ dioptrical work
101
Stroud, Minute, 20; Prins, “Hobbes on light and vision”, 129-132. On Hobbes’ derivation of the sine
law, see section 5.2.1.
102
Lohne, ”Geschichte des Brechungsgesetzes”, 166.
103
Huygens worked on it in 1671-2, see page 160.