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
evolution of the curve ENC, with different lengths of thread, will cut
all the rays HL, GM, FO, etc., at right angles, and in such wise that
the parts of them intercepted between two such curves will all be
equal; for this follows from what has been demonstrated in our
treatise _de Motu Pendulorum_. Now imagining the incident rays as
being infinitely near to one another, if we consider two of them, as
RG, TF, and draw GQ perpendicular to RG, and if we suppose the curve
FS which intersects GM at P to have been described by evolution from
the curve NC, beginning at F, as far as which the thread is supposed
to extend, we may assume the small piece FP as a straight line
perpendicular to the ray GM, and similarly the arc GF as a straight
line. But GM being the refraction of the ray RG, and FP being
perpendicular to it, QF must be to GP as 3 to 2, that is to say in the
proportion of the refraction; as was shown above in explaining the
discovery of Des Cartes. And the same thing occurs in all the small
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