Lenses and Waves

70 CHAPTER 3
the space on the axis within which all parallel rays are brought together,
which space DE is defined by this rule.” 70 Or, the aberration DE is found by
2 2
7n 6an27a multiplying the thickness of the lens q by the expression 2 ,
6(
an) which only depends on the radii of both faces. The shape of a lens that
produces minimal aberration can be found by determining the minimum of
this expression; this yields a : n = 1 : 6. 71 In this case the aberration of the
extreme ray DE = 15q. Huygens found the same for a bi-concave lens,
14
whereas a converging meniscus lens yielded a meaningless outcome. 72
Satisfied, he summarized the result:
“In the optimal lens the radius of the convex objective side is to the radius of the
convex interior side as 1 to 6. EUPHKA. 6 Aug. 1665.” 73
The ‘Adversaria’ provided general expressions for spherical aberration in
terms of the shape of a lens. It contained a set of derivations and calculations
without explanation. He did not, for example, point at certain simplifications
he had carried out. The results were not therefore fully exact, as will become
clear later on. Still, it was the most advanced account of spherical aberration
at the time. On the basis of his theory of spherical aberration he went on to
design a configuration of lenses that minimized the ‘aberrations from the
focus’.
A note of clarification needs to be made. Huygens did not yet call the
phenomenon he was investigating spherical aberration. Around 1665,
Huygens referred to it in a general way: “aberration from the focus” and
“Investigate which convex spherical lens brings parallel rays better
together.” 74 Only much later, when distinguishing the aberration caused by
colors, did he explicitly called it “the aberrations of rays that arise from the
spherical shape of the surfaces”. 75 We should bear this in mind when
interpreting Huygens’ study of aberrations and his designs for perfect
telescopes. That is, we do not know for certain what exactly he thought his
design would improve.
Specilla circularia
Before continuing with Huygens, mention has to be made of another study
of spherical aberration. Not because it mattered much for the mathematical
theory of spherical aberration – it did not – but because it approached the
70 OC13, 364. “DE spatium in axe intra quod radij omnes paralleli coguntur, quod spatium DE per regulam
hanc definitur.”
71 OC13, 366-367. Modern methods yield the same result.
72 OC13, 375 and 370. In the latter case the solution yields a negative value for the radius of the posterior
side.
73 OC13, 367. “Radius convexi objectivi ad radium convexi interioris in lente optima ut 1 ad 6. EUPHKA. 6
Aug. 1665.”
74 OC13, 280n2. “Quaenam lens sphaerica convexa melius radios parallelos coligat investigare.”
75 OC13, 280-281. “aberrationes radiorum quae ex figura superficierum sphaerica oriuntur”