Lenses and Waves

164 CHAPTER 5
In a drawing plus text right above this, the same case is considered, but
now in terms of a wave. 17 (Figure 60) In other words, all rays intersecting in P
and all the points H of intersection of refracted rays are considered. A wave
VK propagates from the glass above VM into the air below it. It propagates in
such a way that all incident rays VP, CP and KP – like the rays AP and DP
above – intersect in P. These rays are refracted towards VM, CG and MP. 18 The
intersections of the refracted rays – like point H above – form a curve VHN
tangent to all refracted rays. Huygens now wanted to prove VHN = NM
3 + 2 MK. That is, the time for light to cover VHN in air, is equal to the time
required to cover NM in air and MK in glass. He explained the meaning of
this statement by considering the moment when point K of the wave has
reached the refracting surface. In the time K moves to M through glass, point
3
V moves to Q through the air (VQ = 2 KM). At this moment, Huygens said
without explanation, a wave is formed consisting of parts RM and RQ, which
are the involutes of parts VR and NR of curve VHN. Ergo, NR + RV equals
3
NM + VQ = NM + 2 MK. It is still not clear what Huygens exactly was after.
He was thinking in terms of rays being paths covered by light in a certain
time, but the point of considering the unfolding wave QR-RM after refraction
is unclear.
Figure 61 A wave refracted at the plane surface of a glass
medium. (Letters ABCDE added by editors, Xxxxx by me)
17 OC19, 421 §2.
18 KP is perpendicular and not refracted; VP is incident at about 48 o, the critical angle, and is refracted to
the parallel VM. It is an odd case, a wave propagating to a single point instead of away from it. It might be
connected to the preceding discussion of ovals, in that Huygens is considering what happens when the
wave has passed the aplanatic surface and crosses a plane surface.