[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 unable to penetrate into the other transparent substance.  For if the angle DAQ or CBA is such that in the triangle ACB, CB is equal to 2/3 of ab, or is greater, then an cannot form one side of the triangle ANB, since it becomes equal to or greater than ab:  so that the portion of wave BN cannot be found anywhere, neither consequently can an, which ought to be perpendicular to it.  And thus the incident ray DA does not then pierce the surface ab.

When the ratio of the velocities of the waves is as two to three, as in our example, which is that which obtains for glass and air, the angle DAQ must be more than 48 degrees 11 minutes in order that the ray DA may be able to pass by refraction.  And when the ratio of the velocities is as 3 to 4, as it is very nearly in water and air, this angle DAQ must exceed 41 degrees 24 minutes.  And this accords perfectly with experiment.

But it might here be asked:  since the meeting of the wave AC against the surface ab ought to produce movement in the matter which is on the other side, why does no light pass there?  To which the reply is easy if one remembers what has been said before.  For although it generates an infinitude of partial waves in the matter which is at the other side of ab, these waves never have a common tangent line (either straight or curved) at the same moment; and so there is no line terminating the propagation of the wave AC beyond the plane ab, nor any place where the movement is gathered together in sufficiently great quantity to produce light.  And one will easily see the truth of this, namely that CB being larger than 2/3 of ab, the waves excited beyond the plane ab will have no common tangent if about the centres K one then draws circles having radii equal to 3/2 of the lengths lb to which they correspond.  For all these circles will be enclosed in one another and will all pass beyond the point B.

Now it is to be remarked that from the moment when the angle DAQ is smaller than is requisite to permit the refracted ray DA to pass into the other transparent substance, one finds that the interior reflexion which occurs at the surface ab is much augmented in brightness, as is easy to realize by experiment with a triangular prism; and for this our theory can afford this reason.  When the angle DAQ is still large enough to enable the ray DA to pass, it is evident that the light from the portion AC of the wave is collected in a minimum space when it reaches BN.  It appears also that the wave BN becomes so much the smaller as the angle CBA or DAQ is made less; until when the latter is diminished to the limit indicated a little previously, this wave BN is collected together