But he assumed besides this a constant ratio of Sines, which we have
just proved by these different degrees of velocity alone: or rather,
what is equivalent, he assumed not only that the velocities were
different but that the light took the least time possible for its
passage, and thence deduced the constant ratio of the Sines. His
demonstration, which may be seen in his printed works, and in the
volume of letters of Mr. Des Cartes, is very long; wherefore I give
here another which is simpler and easier.

[Illustration]

Let KF be the plane surface; A the point in the medium which the light
traverses more easily, as the air; C the point in the other which is
more difficult to penetrate, as water. And suppose that a ray has come
from A, by B, to C, having been refracted at B according to the law
demonstrated a little before; that is to say that, having drawn PBQ,
which cuts the plane at right angles, let the sine of the angle ABP
have to the sine of the angle CBQ the same ratio as the velocity of
light in the medium where A is to the velocity of light in the medium
where C is. It is to be shown that the time of passage of light along
AB and BC taken together, is the shortest that can be. Let us assume
that it may have come by other lines, and, in the first place, along
AF, FC, so that the point of refraction F may be further from B than
the point A; and let AO be a line perpendicular to AB, and FO parallel
to AB; BH perpendicular to FO, and FG to BC.

Since then the angle HBF is equal to PBA, and the angle BFG equal to
QBC, it follows that the sine of the angle HBF will also have the same
ratio to the sine of BFG, as the velocity of light in the medium A is
to its velocity in the medium C. But these sines are the straight