Problem 3. Given that __/__AOE = 38^{o}, O is the
center of the circle, NO__ | __OE, NO || AP, and NE is tangent to
the circle at A,

.find __/__NAP

Since PB is parallel to ON and On is perpendicular to OE, PB is
perpendicular to OE. This gives us two angles in triangle AOB, and
one of them is a right angle. So the other angle, __/__OAB is the
complement of 38^{o} or 52^{o}.

Since NE is tangent to the circle at A and since radii are
perpendicular to tangents, it follows that __/__BAE is the
complement of __/__AOB.

And, finally, /NAP is a vertical angle to /BAE

If the picture represents the Earth, and NS is the Earth's axis, then AP, being parallel to ON is the direction where the North Star, Polaris, appears. In that case, /NAP is the angle above the horizon where a person standing at point A would see Polaris. /AOE is the lattitude of point A. This shows that the angle above the horizon where the North Star appears is the same as the lattitude of the point A. Using this method, people even in days of the greatest antiquity could tell that the Earth was not flat.