t FD up to the straight line VG in
such a way that FD together with 3/2 of DG is equal to a given
straight line, which is a quite easy plane problem: and the point D
will be one of those through which the curve BDK ought to pass. And
similarly, having drawn another ray LM, and found its refraction MO,
the point N will be found in this line, and so on as many times as one
desires.

To demonstrate the effect of the curve, let there be described about
the centre L the circular arc AH, cutting LG at H; and about the
centre F the arc BP; and in AB let AS be taken equal to 2/3 of HG; and
SE equal to GD. Then considering AH as a wave of light emanating from
the point L, it is certain that during the time in which its piece H
arrives at G the piece A will have advanced within the transparent
body only along AS; for I suppose, as above, the proportion of the
refraction to be as 3 to 2. Now we know that the piece of wave which
is incident on G, advances thence along the line GD, since GV is the
refraction of the ray LG. Then during the time that this piece of wave
has taken from G to D, the other piece which was at S has reached E,
since GD, SE are equal. But while the latter will advance from E to B,
the piece of wave which was at D will have spread into the air its
partial wave, the semi-diameter of which, DC (supposing this wave to
cut the line DF at C), will be 3/2 of EB, since the velocity of light
outside the medium is to that inside as 3 to 2. Now it is easy to show
that this wave will touch the arc BP at this point C. For since, by
construction, FD + 3/2 DG + GL are equal to FB + 3/2 BA + AL; on
deducting the equals LH, LA, there will remain FD + 3/2 DG + GH equal
to FB + 3/2 BA. And, again, deducting from one side GH, and from the
other side 3/2 of AS, which are equal, there will remain FD with 3/2
DG equal to FB with 3/2 of BS. But 3/2 of DG are equal to 3/2 of ES;
then FD is equal to FB with 3/2 of BE. But DC was equal to 3/2 of EB;
then deducting these equal lengths from one side and from the other,
there will remain CF equal to FB. And thus it appears that the wave,
the semi-diameter of which is DC, touches the arc BP at the moment
when the light coming from the point L has arrived at B along the line
LB. It can be demonstrated similarly that at this same moment the
light that has come along any other ray, such as LM, MN, will have
propagated the movement which is terminated at the arc BP. Whence it
follows, as has