  For millenia, sailors have used the empirical rule that the elevation angle
of Polaris, the North Star, as measured by sextant, quadrant or astrolabe, is
approximately equal to latitude. Here, we show using elementary trigonometry
that Empirical Law 1 can be converted from a heuristic to a theorem. A second
ancient empirical law is that the distance in kilometers from the observer to
the North Pole, the geodesic distance measured along the spherical surface of
the planet, is the number of degrees of colatitude multiplied by 111.1
kilometers. Can Empirical Law 2 be similarly rendered rigorous? No; whereas as
the shape of the planet is controlled by trigonometry, the size of our world is
an accident of cosmological history. However, Empirical Law 2, can be
rigorously verified by measurements. The association of 111 km of north-south
distance to one degree of latitude trivially yields the circumference of the
globe as 40,000 km. We also extend these ideas and the parallel ray
approximation to three different ways of modeling a Flat Earth. We show that
photographs from orbit, taken by a very expensive satellite, are unnecessary to
render the Flat Earth untenable; simple mathematics proves Earth a sphere just
as well.
