It is then BN, which is formed by small arcs of these circumferences,
which terminates the movement that the wave AC has communicated within
the transparent body, and where this movement occurs in much greater
amount than anywhere else. And for that reason this line, in
accordance with what has been said more than once, is the propagation
of the wave AC at the moment when its piece C has reached B. For there
is no other line below the plane AB which is, like BN, a common
tangent to all these partial waves. And if one would know how the wave
AC has come progressively to BN, it is necessary only to draw in the
same figure the straight lines KO parallel to BN, and all the lines KL
parallel to AC. Thus one will see that the wave CA, from being a
straight line, has become broken in all the positions LKO
successively, and that it has again become a straight line at BN. This
being evident by what has already been demonstrated, there is no need
to explain it further.
Now, in the same figure, if one draws EAF, which cuts the plane AB at
right angles at the point A, since AD is perpendicular to the wave AC,
it will be DA which will mark the ray of incident light, and AN which
was perpendicular to BN, the refracted ray: since the rays are nothing
else than the straight lines along which the portions of the waves
advance.
Whence it is easy to recognize this chief property of refraction,
namely that the Sine of the angle DAE has always the same ratio to the
Sine of the angle NAF, whatever be the inclination of the ray DA: and
that this ratio is the same as that of the velocity of the waves in
the transparent substance which is towards AE to their velocity in the
transparent substance towards AF. For, considering AB as the radius of
a circle, the Sine of the angle BAC is BC, and the Sine of the angle
ABN is AN. But the angle BAC is equal to DAE, since each of them added
to CAE makes a right angle. And the angle ABN is equal to NAF, since
each of them with BAN makes a right angle. Then also the Sine of the
angle DAE is to the Sine of NAF as BC is to AN. But the ratio of BC to
AN was the same as that of the velocities of light in the substance
which is towards AE and in that which is towards AF; therefore also
the Sine of the angle DAE will be to the Sine of the angle NAF the
same as the said velocities of light.
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