91Ó°ÊÓ

Use al-B?r?ni's procedure to determine the qibla for Rome (latitude \(41^{\circ} 53^{\prime} \mathrm{N}\), longitude \(12^{\circ} 30^{\prime} \mathrm{E}\) ).

Show that the radius \(r_{\alpha}\) of a latitude circle on the earth at \(\alpha^{\circ}\) is given by \(r_{\alpha}=R \cos \alpha\), where \(R\) is the radius of the -earth.

The latitudes of Philadelphia and Ankara, Turkey, are the same \(\left(40^{\circ}\right)\), with the first at longitude \(75^{\circ} \mathrm{W}\) and the second at longitude \(33^{\circ} \mathrm{E}\). Calculate the distance between Philadelphia and Ankara along the latitude circle, by first calculating the radius of that circle, using 25,000 miles for the circumference of the earth. Then calculate the distance along a great circle, by noting that the chord connecting the two cities can be thought of as a chord of that circle as well as a chord of the latitude circle. (Hint: You will have to convert the chords to the appropriate sines to make this calculation.)

Al-B?r?ni devised a method for determining the radius \(r\) of the earth by sighting the horizon from the top of a mountain of known height \(h\). That is, al-B?r?ni assumed that one could measure \(\alpha\), the angle of depression from the horizontal at which one sights the apparent horizon (Fig. 9.38). Show that \(r\) is determined by the formula $$ r=\frac{h \cos \alpha}{1-\cos \alpha} $$ Al-B?r?n? performed this measurement in a particular case, determining that \(\alpha=0^{\circ} 34^{\prime}\) as measured from the summit of a mountain of height \(652 ; 3,18\) cubits. Calculate the radius of the earth in cubits. Assuming that a cubit equals \(18^{\prime \prime}\), convert your answer to miles and compare to a modern value. Comment on the efficacy of al-Bir?ni's procedure.

Use al-T?si's method to solve the spherical triangle with known sides of \(40^{\circ}\) and \(50^{\circ}\) and with the angle between those sides equal to \(25^{\circ}\).

Why did it take many centuries after its introduction for the decimal place value system to become the system of numeration universally used in the Islamic world?