simple enough in theory. Its defect consists largely in the
difficulty of fixing its terms with precision, combined with the
further fact that the rays of the sun, in taking the slanting
course through the earth's atmosphere, are really deflected from
a straight line in virtue of the constantly increasing density of
the air near the earth's surface. Alhazen must have been aware of
this latter fact, since it was known to the later Alexandrian
astronomers, but he takes no account of it in the present
measurement. The diagram will make the method of Alhazen clear.

His important premises are two: first, the well-recognized fact
that, when light is reflected from any surface, the angle of
incidence is equal to the angle of reflection; and, second, the
much more doubtful observation that twilight continues until such
time as the sun, according to a simple calculation, is nineteen
degrees below the horizon. Referring to the diagram, let the
inner circle represent the earth's surface, the outer circle the
limits of the atmosphere, C being the earth's centre, and RR
radii of the earth. Then the observer at the point A will
continue to receive the reflected rays of the sun until that body
reaches the point S, which is, according to the hypothesis,
nineteen degrees below the horizon line of the observer at A.
This horizon line, being represented by AH, and the sun's ray by
SM, the angle HMS is an angle of nineteen degrees. The
complementary angle SMA is, obviously, an angle of (180-19) one
hundred and sixty-one degrees. But since M is the reflecting
surface and the angle of incidence equals the angle of
reflection, the angle AMC is an angle of one-half of one hundred
and sixty-one degrees, or eighty degrees and thirty minutes. Now
this angle AMC, being known, the right-angled triangle MAC is