Lenses and Waves

1655-1672 - DE ABERRATIONE 81
the other way around. 126 The bi-concave lens VBC is given. M is the center of
surface BV, so that rays from M are not refracted by it. Surface CB of this lens
refracts a ray MC to KCN, intersecting the axis in N, where EN is the spherical
aberration. The problem is to find a convex lens KST with the same focus E,
which refracts a parallel ray QK, at distance KS to the axis, to the same point
N. Huygens chose BE – nearly equal to GE – as the focal distance of this lens
KST. Its spherical aberration EN is – by the rule from the ‘Adversaria’ – 7
6
times its thickness GS. This length EN is also the spherical aberration of
surface BC of the bi-concave lens VBC. It can be expressed in terms of its
radius AB, the length BG (proportional to the distance CG of the ray to the
axis), and the length BM. Equating both expressions for EN, he found a
proportionality between the radius of KST and BC. It is 100 to 254, or nearly
2 to 5. In addition MB, the radius of the other surface BV of the bi-concave
lens, has to be twice that of BC or ten times that of KST. At the end of his
calculations Huygens summarized the solution:
“A lens composed of two emulates a hyperbolic lens, the one plano-convex the other
concave on both sides. The radii of the surfaces are nearly two, five, ten.” 127
Five days later, on 6 February, he sent a letter to Oldenburg to which he
appended an anagram containing his ‘important invention’: 128
a bc
d e h i l m n op
r s t u y
52
2 14
1 23
3 1 3 2 23
24
1
This second invention can be regarded as the final piece of Huygens’ project
of canceling out spherical aberration by means of spherical lenses. He had
shown that spherical lenses were indeed apt for telescopes by designing a
configuration that produced an almost perfect focus. As contrasted to the
earlier invention of 1665, this one could improve telescopes used for
astronomical observation. 129 What remained to be done, was to test the
design.
We should remember that it was not an ordinary project Huygens had
embarked on. His theoretical investigations of spherical aberration served
the practical goal of improving actual telescopes. With this he marked
himself off from both theoreticians and practitioners. Unlike other telescope
makers – as he manifested himself earlier – he had aimed at improving the
telescope by means of theoretical study. The configuration in which
aberration was to be neutralized was not the result of trial-and-error like his
eyepiece, but of mathematical analysis of lenses and calculating the optimal
126
OC13, 411-413.
127
OC13, 417n2. “Lens e duabus composita hyperbolicam aemulatur, altera planoconvexa altera cava
utrimque. Semidiametri superficierum sunt proximè duo, quinque, decem.”
128
OC4, 354-355 and OC13, 417. The solution of the anagram is: “Lens e duabus composita hyperbolicam
aemulatur”.
129
Huygens may have tested the idea to combine two lenses into an objective earlier, at the time of the
invention of 1665. Hug29, 76v and 77r contain sketches reminiscent of the earlier invention as well as
ones reminiscent of the 1669 invetion. The folios can date from any time between the two inventions, but
appear to reflect some intermediate stage in his thinking.