Lenses and Waves

36 CHAPTER 2
century student of burning glasses Ibn Sahl had used a rule equivalent to the
sine law. 90 Around 1620, the Leiden professor of mathematics Willebrord
Snel was next and in the late 1620s Descartes closed the ranks of discoverers
of the law of refraction. 91 He published it in La Dioptrique (1637), shortly after
Pierre Hérigone had done so in the fifth volume of Cursus Mathematicus.
Hérigone did not use it in his dioptrical account, which summarized Dioptrice.
Harriot and Snel have left no trace of applying their find to lenses. Which
leaves La Dioptrique for further inspection.
Descartes and the ideal telescope
La Dioptrique was the fruit of Descartes’ involvement in the activities of
Parisian savants regarding (non-spherical) mirrors and lenses, which also
places him in the sixteenth-century tradition of mirror-making. 92 Descartes,
however, added his natural philosophical leanings and Kepler’s optical
teachings. In collaboration with the mathematician Mydorge and the artisan
Ferrier, he allegedly managed to produce a hyperbolic lens and in the course
of events he discovered the law of refraction. La Dioptrique had much
influence on seventeenth-century optics, especially through its second
discourse where Descartes derived the sine law. 93 In the following discourses,
Descartes first discussed the eye and vision – summarizing Kepler’s theory
of the retinal image – and then went on to a consideration “Of the means of
perfecting vision”. 94 This seventh discourse anticipated his discussion of
telescopes. He laid stress on the way spectacles enhance vision, instead of
correcting it. The telescope itself was introduced in a peculiar way. Descartes
explained how an elongated lens may further enhance vision. He then
replaced the solid middle part by air, thus arriving at a telescope consisting of
two lenses. 95 The argument was clear, but the discussion of focal and
magnifying properties of lenses was entirely qualitative and the sine law
played no role in it.
In the eighth discourse of La Dioptrique, Descartes applied the sine law to
lenses under the title: “Of the figures transparent bodies must have to divert
the rays by refraction in all manners that serve vision”. 96 Its sole purpose was
to show that lenses ought to have an elliptic or hyperbolic surface in order to
bring rays to a perfect focus. Avoiding the subtleties of geometry he
explained how these lines could be drawn by practical means and
demonstrated the relevant properties of the ellipse and hyperbola. As regards
the focal distances of lenses thus obtained with respect to configuration and
90
Rashed, “Pioneer”, 478-486.
91
For Snel see: Hentschel, “Brechungsgesetz”. It is possible that Wilhelm Boelmans in Louvain somewhat
later discovered the sine law independently. Ziggelaar, “The sine law of refraction”, 250.
92
Gaukroger, Descartes, 138-146. Dupré, Galileo, the Telescope, 53-54.
93
Discussed in section 4.1.3
94
Descartes, AT6, 147. “Des moyens de perfectionner la vision. Discours septiesme.”
95
Descartes, AT6, 155-160.
96
Descartes, AT6, 165. “Des figures que doivent avoir les corps transparens pour detourner les rayons par
refraction en toutes les façons qui servent a la veuë”