n the medium A is
to its velocity in the medium C. But these sines are the straight
lines HF, BG, if we take BF as the semi-diameter of a circle. Then
these lines HF, BG, will bear to one another the said ratio of the
velocities. And, therefore, the time of the light along HF, supposing
that the ray had been OF, would be equal to the time along BG in the
interior of the medium C. But the time along AB is equal to the time
along OH; therefore the time along OF is equal to the time along AB,
BG. Again the time along FC is greater than that along GC; then the
time along OFC will be longer than that along ABC. But AF is longer
than OF, then the time along AFC will by just so much more exceed the
time along ABC.

Now let us assume that the ray has come from A to C along AK, KC; the
point of refraction K being nearer to A than the point B is; and let
CN be the perpendicular upon BC, KN parallel to BC: BM perpendicular
upon KN, and KL upon BA.

Here BL and KM are the sines of angles BKL, KBM; that is to say, of
the angles PBA, QBC; and therefore they are to one another as the
velocity of light in the medium A is to the velocity in the medium C.
Then the time along LB is equal to the time along KM; and since the
time along BC is equal to the time along MN, the time along LBC will
be equal to the time along KMN. But the time along AK is longer than
that along AL: hence the time along AKN is longer than that along ABC.
And KC being longer than KN, the time along AKC will exceed, by as
much more, the time along ABC. Hence it appears that the time along
ABC is the shortest possible; which was to be proven.

CHAPTER IV

ON THE REFRACTION OF THE AIR

We have shown how the movement which constitutes light spreads by
spherical waves in any homogeneous matter. And it is evident that when
the matter is not homogeneous, but of such a constitution that the
movement is communicated in it more rapidly toward one side than
toward another, these waves cannot be spherical: but that they must
acquire their figure according to the different distances over which
the successive movement passes in equal times.

It is thus that we shall in the first place explain the refractions
which occur in the air, which extends from here to the clouds and
beyond. The effects of which refractions are very remarkable; for by
them we often see objects which the rotundity of the Earth ought
otherwise to hide; such as Islands, and the tops of mountain