the arc RX at the moment when the piece Q shall
have come to R, and that thus this arc will at the same moment be the
termination of the movement that comes from the wave TG; whence all
the rest may be concluded.

Having shown the method of finding these curved lines which serve for
the perfect concurrence of the rays, there remains to be explained a
notable thing touching the uncoordinated refraction of spherical,
plane, and other surfaces: an effect which if ignored might cause some
doubt concerning what we have several times said, that rays of light
are straight lines which intersect at right angles the waves which
travel along them.

[Illustration]

For in the case of rays which, for example, fall parallel upon a
spherical surface AFE, intersecting one another, after refraction, at
different points, as this figure represents; what can the waves of
light be, in this transparent body, which are cut at right angles by
the converging rays? For they can not be spherical. And what will
these waves become after the said rays begin to intersect one another?
It will be seen in the solution of this difficulty that something very
remarkable comes to pass herein, and that the waves do not cease to
persist though they do not continue entire, as when they cross the
glasses designed according to the construction we have seen.

According to what has been shown above, the straight line AD, which
has been drawn at the summit of the sphere, at right angles to the
axis parallel to which the rays come, represents the wave of light;
and in the time taken by its piece D to reach the spherical surface
AGE at E, its other parts will have met the same surface at F, G, H,
etc., and will have also formed spherical partial waves of which these
points are the centres. And the surface EK which all those waves will
touch, will be the continuation of the wave AD in the sphere at the
moment when the piece D has reached E. Now the line EK is not an arc
of a circle, but is a curved line formed as the evolute of another
curve ENC, which touches all the rays HL, GM, FO, etc., that are the
refractions of the parallel rays, if we imagine laid over the
convexity ENC a thread which in unwinding describes at its end E the
said curve EK. For, supposing that this curve has been thus described,
we will show that the said waves formed from the centres F, G, H,
etc., will all touch it.

It is certain that the curve EK and all the others described by the
ev