91Ó°ÊÓ

Use Ptolemy's difference formula to calculate \(\operatorname{crd}\left(12^{\circ}\right)\) and then apply the half-angle formula to calculate crd \(\left(6^{\circ}\right)\), \(\operatorname{crd}\left(3^{\circ}\right), \operatorname{crd}\left(1 \frac{1}{2}^{\circ}\right)\), and \(\operatorname{crd}\left(\frac{3}{4}\right)\). Compare your results to Ptolemy's.

Calculate the declination and right ascension of the sun when it is at longitude \(90^{\circ}\) (summer solstice) and longitude \(45^{\circ}\). By symmetry, find the declination at longitudes \(270^{\circ}\) and \(315^{\circ}\).

. At approximately what dates is the sun directly overhead at noon at a place whose geographical latitude is \(20^{\circ}\) ?

Show how to calculate the distance between two inaccessible points \(A, B\), by the use of similar triangles. (Assume, for example, that the two points are on the bank of a river opposite your position.)

. Show that the total length of the parallel at latitude \(\alpha\) equals \(\cos \alpha\) multiplied by the total length of the equator.

List evidence that convinces you that the earth (a) rotates on its axis once a day and (b) revolves around the sun once a year. Would this evidence have convinced the Greeks? How would you refute the reasons Ptolemy gives for the earth's immovability?