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 mountains when one
is at sea. Because also of them the Sun and the Moon appear as risen
before in fact they have, and appear to set later: so that at times
the Moon has been seen eclipsed while the Sun appeared still above the
horizon. And so also the heights of the Sun and of the Moon, and those
of all the Stars always appear a little greater than they are in
reality, because of these same refractions, as Astronomers know. But
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