Chapter 5

Calculate crd \(\left(30^{\circ}\right), \operatorname{crd}\left(15^{\circ}\right)\), and \(\operatorname{crd}\left(7 \frac{1}{2}^{\circ}\right)\) using the halfangle tormula of Hipparchus, beginning with the fact that \(\operatorname{cod}\left(60^{2}\right)=R=60 .\)

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.

Prove that \(\operatorname{crd} \beta\) : crd \(\alpha<\beta: \alpha\) or, equivalently, that \(\frac{\sin \beta}{\sin \alpha}<\) \(\frac{\beta}{\alpha}\) for \(0<\alpha<\beta\)

calculate, using Ptolemy's methods, the length of a noon. shadow of a pole of length 60 at the vernal equinox at a place of latitude \(40^{\circ}\).

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 \(315^{\circ}\).

Calculate the rising times \(\rho(\lambda, \phi)\) for \(\phi=45^{\circ}\) and \(\lambda=60^{\circ}\) and \(90^{\circ}\)

Suppose that the maximum length of day at a particular location is known to be 15 hours. Calculate the latitude of

The formula \(\sin \sigma=\tan \delta \tan \phi\) only makes sense if the right- hand side is less than or equal to I. Since the maximum value of \(\delta\) is \(23 \frac{1}{2}^{\circ}\), show that the right-hand side will be greater than 1 whenever \(\phi>66 \frac{1^{\circ}}{2} .\) Interpret the formula in this case in terms of the length of daylight.

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

Compare the formula \(A=\left(\frac{1}{3}+\frac{1}{10}\right) s^{2}\) for the area of an equilateral triangle of side \(s\), used by a Roman surveyor, with the exact formula. What approximation has the surveyor used for \(\sqrt{3} ?\)

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.)

Derive a formula for the area \(A_{5}\) of a regular pentagon, with side \(a\) (using plane geometry). Discuss the differences between Heron's formula \(A_{5}=\frac{5}{3} a^{2}\) and your formula.

Heron derived his formula for the area \(A_{7}\) of a regular heptagon of side \(a, A_{7}=\frac{43}{12} a^{2}\), by assuming that \(a=\frac{7}{8} r\) where \(r\) is the radius of the circumscribed circle. Use this approximation to derive Heron's result. What square root approximation is necessary here?

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

Ptolemy must have been aware of a method of trisecting angles by the use of conic sections. Such a method would have enabled him to construct the chord of \(\frac{1}{2}^{\circ}\) given that he knew the chord of \(1 \frac{1}{2}^{\circ}\). Why would Ptolemy not have considered this to be a construction by "geometrical methods"? Can one use such a construction to calculate the chord of \(\frac{1}{2}^{\circ}\) numerically?

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?