Lenses and Waves

1655-1672 - DE ABERRATIONE 73
1 ( 1000000 inches) but this was of no significance in actual telescopes. 87 When
the convex side of the same lens faced the incident rays, the exact calculation
yielded an aberration of 81021 inches. In this case, the easier rule of
10000000
‘Adversaria’ – multiplying the thickness of the lens by 7
6 – gave 81022
10000000 , a
1
difference of only inches. Again, the main goal of this exercise was to
1000000
show that the aberration of a plano-convex lens is least when its convex side
faces the incident rays. 88 Continuing with a bi-convex lens, Huygens sketched
out how the aberration might be calculated exactly, but immediately moved
on to an ‘abbreviated rule’ he had ‘found’. 89 This was the expression of the
‘Adversaria’, found by “ignoring very little quantities, but judiciously so as
needed.” 90 The rule applied to convex as well as to concave lenses and
yielded the optimal proportion of both radii of 1 : 6. The resulting bi-convex
15 91
times its thickness.
lens produces an aberration of only 14
Surprisingly, these laborious derivations were not of great value for
telescopes. After having explained the optimal proportions of bi-concave
lenses, Huygens wrote that they were not useful as ocular lenses. In
telescopes, he said, one should choose “… other, less perfect lenses, so that
the defects of the convex lens are compensated and corrected by their
defects.” 92 Those less perfect lenses were diverging concavo-convex lenses.
Huygens showed that these lenses always produce a larger aberration than biconvex
or bi-concave lenses.
As ocular lenses they could, however, be useful:
“With concave and convex spherical lenses, to make telescopes that are better than the
one made according to what we know now, and that emulate the perfection of those
that are made with elliptic or hyperbolic lenses.” 93
Here was what Huygens had been looking for: a configuration where lenses
mutually cancel out their aberration. He had designed a telescope in which
the ocular corrects for the aberration of the objective lens, thus equaling the
effect of a-spherical lenses.
The solution was as follows: given an objective lens and the required
magnification of a telescope, determine the shape of the ocular lens (Figure
29). On the axis BDFE of lens ABCD, divide the focal distance DE by point F
87
OC13, 284-285. “Exigua quidem differentiola, sed quae in illa lentium latitudine quae telescopiorum
usibus idonea est, nullius sit momenti.”
88
OC13, 284-287.
89
OC13, 290-291. “Et haec quidem methodus ad exactam supputationem adhibenda esset. Invenimus
autem et hic Regulam compendiosam …”
90
OC13, 290-291. “Quae regula … inventa est neglectis minimis, sed necessario cum delectu.”
91
OC13, 290-291& 302-303.
92
OC13, 302-303. “…, sed aliae minus perfectae, quarum nempe vitijs compensantur ac corrigentur vitia
lentis convexae, …”
93
OC13, 318-319. “Ex lentibus sphæricis cavis et convexis telesopia componere hactenus cognitis ejus
generis meliora, perfectionemque eorum quæ ellipticis hyperbolicisve lentibus constant æmulantia.”