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.

To see, consequently, what the refraction will be when the waves of
light pass into a substance in which the movement travels more quickly
than in that from which they emerge (let us again assume the ratio of
3 to 2), it is only necessary to repeat all the same construction and
demonstration which we have just used, merely substituting everywhere
3/2 instead of 2/3. And it will be found by the same reasoning, in
this other figure, that when the piece C of the wave AC shall have
reached the surface AB at B, all the portions of the wave AC will
have advanced as far as BN, so that BC the perpendicular on AC is to
AN the perpendicular on BN as 2 to 3. And there will finally be this
same ratio of 2 to 3 between the Sine of the angle BAD and the Sine of
the angle FAN.

Hence one sees the reciprocal relation of the refractions of the ray
on entering and on leaving one and the same transparent body: namely
that if NA falling on the external surface AB is refracted into the
direction AD, so the ray AD will be refracted on leaving the
transparent body into the direction AN.

[Illustration]

One sees also the reason for a noteworthy accident which happens in
this refraction: which is this, that after a certain obliquity of the
incident ray DA, it begins to be quite unab