been often said, that the propagation of the wave AH,
after it has passed through the thickness of the glass, will be the
spherical wave BP, all the pieces of which ought to advance along
straight lines, which are the rays of light, to the centre F. Which
was to be proved. Similarly these curved lines can be found in all the
cases which can be proposed, as will be sufficiently shown by one or
two examples which I will add.

Let there be given the surface of the glass AK, made by the revolution
about the axis BA of the line AK, which may be straight or curved. Let
there be also given in the axis the point L and the thickness BA of
the glass; and let it be required to find the other surface KDB, which
receiving rays that are parallel to AB will direct them in such wise
that after being again refracted at the given surface AK they will all
be reassembled at the point L.

[Illustration]

From the point L let there be drawn to some point of the given line
AK the straight line LG, which, being considered as a ray of light,
its refraction GD will then be found. And this line being then
prolonged at one side or the other will meet the straight line BL, as
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 desire