Lenses and Waves

1677-1679 –WAVES OF LIGHT 163
second point. 13 (In a way this brought him back to the very beginning of his
dioptrical studies.) Then he returned to the principle of least time, now to
derive it from the sine law (Fermat had worked the other way around). 14 All
of this dealt with rays refracted by plane and curved surfaces.
Figure 59 Two rays refracted by a plane
surface. Lettering added.
Figure 60 Wave VK refracted by a plane surface
VM forming a caustic VHN.
On the next page, Huygens moved on to the case when rays do not
intersect in one point after refraction. 15 In such cases, the intersections of
refracted rays form a caustic. Huygens first considered two rays refracted
from glass (top) to air (bottom) at a plane surface (Figure 59). Before
refraction, the rays AP and DP intersect in P. AP : AE = DP : DF = 3 : 2,
according to the sine law. The refracted rays AE and DF intersect in H.
3 Huygens continued: “the difference of the two PA, PD must be 2 of the
difference of the two EA, FD.” 16 These differences are AB and AC
3 respectively; AC equals 2 AB. They indicate the paths traversed by rays in air
and glass in equal times: in the time light covers the distance AC in air, its
covers AB in glass. With this, Huygens derived an expression for the position
of H on AE in terms of the position of S (OS perpendicular to AE).
13
Hug9, 38v-40r; partly reproduced in §1and §2 on OC19, 424-425.
14
Hug9, 40v; §2 on OC19, 416-417.
15
Hug9, 41r. This page (reproduced in OC19, 419 §2) begins with an analysis, invoking his theory of
spherical aberration, of the point of intersection of two near parallel rays refracted by a sphere.
16
OC19, 418. “differentia duarum PA, PD debet esse 3/2 differentiae duarum EA, DF.”