Lenses and Waves

24 CHAPTER 2
system, Huygens applied
proposition six of book
two. The eye is imagined
at S and the object at PN.
In this way the
magnification is
determined by the Figure 13 Diagram for the analysis in Figure 12.
proportion YB : PN. It easily follows that this proportion is equal to AC : EL,
the proportion of the focal distances of the objective and the ocular.
Conclusion
In Tractatus, Huygens addressed a specific question: how can the working of
the telescope be understood mathematically? Regarding thin glass lenses his
answers, as we shall see in the next section, were not that new. Yet, he had
arrived at these answers by way of a rigorous mathematical analysis of the
properties of lenses. With the sine law, Huygens derived general and exact
theorems regarding the focal distances of thick lenses for both parallel and
non-parallel rays, irrespective of the material lenses are made of. On the basis
of this exact theory, he showed that these theorems reduce to the familiar,
simpler ones when the thickness of the lens is ignored and a specific index of
refraction is chosen. In the same way, he first established a general theorem
regarding the magnification by a lens-system and then showed that, in the
cases of actual telescopes, it reduced to the simple and familiar one. If the
elaboration of the theory of Tractatus was markedly mathematical, its
rationale was the telescope. Its goal was a ‘theory of the telescope’: an
account of the working of the telescope on the basis of dioptrical theory. In
this sense, the theory of the first two books was almost too elaborate. All in
all, in his Tractatus, Huygens gave a rigorous answer to the question how the
working of the telescope can be understood mathematically.
Huygens was the first one to elaborate a theory of the teleoscope by
means of the exact law of refraction. He knew that his treatise would fill gaps
left by others, in particular Descartes, so we would expect him to publish it
soon. However, as contrasted to other mathematical treatises he published in
this period, he did not press ahead with Tractatus. He inquired with
publishers and Van Schooten even proposed to append Huygens’ treatise to
a Latin edition of Descartes’ Discours de la Methode, La Dioptrique and Les
Météores, but nothing came of it. 43 Despite repeated announcements between
1655 and 1665 that he was publishing Tractatus, Huygens never did. 44
2.2 Dioptrics and the telescope
The orientation on the telescope is essential to Tractatus. If Huygens was the
first to apply the sine law to questions regarding the telescope, what had
other students of dioptrics been doing? In this section, I sketch the
43
OC1, 280; 301-303; 321-322. Huygens did not pin much faith in Van Schooten’s proposal.
44
I will say a bit more about his publishing pattern on page 174.