here at V. Let there then be erected on AB the perpendicular BC, which
will represent a wave of light coming from the infinitely distant
point F, since we have supposed the rays to be parallel. Then all the
parts of this wave BC must arrive at the same time at the point L; or
rather all the parts of a wave emanating from the point L must arrive
at the same time at the straight line BC. And for that, it is
necessary to find in the line VGD the point D such that having drawn
DC parallel to AB, the sum of CD, plus 3/2 of DG, plus GL may be equal
to 3/2 of AB, plus AL: or rather, on deducting from both sides GL,
which is given, CD plus 3/2 of DG must be equal to a given length;
which is a still easier problem than the preceding construction. The
point D thus found will be one of those through which the curve ought
to pass; and the proof will be the same as before. And by this it will
be proved that the waves which come from the point L, after having
passed through the glass KAKB, will take the form of straight lines,
as BC; which is the same thing as saying that the rays will become
parallel. Whence it follows reciprocally that parallel rays falling on
the surface KDB will be reassembled at the point L.
[Illustration]
Again, let there be given the surface AK, of any desired form,
generated by revolution about the axis AB, and let the thickness of
the glass at the middle be AB. Also let the point L be given in the
axis behind the glass; and let it be supposed that the rays which fall
on the surface AK tend to this point, and that it is required to find
the surface BD, which on their emergence from the glass turns them as
if they came from the point F in front of the glass.
Having taken any point G in the line AK, and drawing the straight line
IGL, its part GI will represent one of the incident rays, the
refraction of which, GV, will then be found: and it is in this line
that we must find the point D, one of those through which the curve DG
ought to pass. Let us suppose that it has been found: and about L as
centre let there be described GT, the arc of a circle cutting the
straight line AB at T, in case the distance LG is greater than LA; for
otherwise the arc AH must be described about the same centre, cutting
the straight line LG at H. This arc GT (or AH, in the other case) will
represent an incident wave of light, the rays of which tend towards
L. Similarly, about the centre F let there be described the circular
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