268 9. MEASUREMENTCompare this list with Aryabhata's list, and note the systematic divergence.
These differences should be approximately 225 times the cosine of the appropriate
angle. That is, dn « 225 · cos (225(n + 0.5) minutes). What does that fact suggest
about the source of the systematic errors in the recursive procedure described by
Aryabhata?9.17. Use Aryabhata's procedure to compute the altitude of the Sun above the
horizon in London (latitude 51° 32') at 10:00 AM on the vernal equinox. Assume
that the sun rises at 6:00 AM on that day and sets at 6:00 PM.9.18. Why is it necessary that a quadrilateral be inscribed in a circle in order to
compute its diagonals knowing the lengths of its sides? Why is it not possible to
do so in general?
9.19. Show that the formula given by Brahmagupta for the area of a quadrilateral
is correct if and only if the quadrilateral can be inscribed in a circle.
9.20. Imagine a sphere as a polyhedron having a large number of very small faces.
Deduce the relation between the volume of a sphere and its area by considering the
pyramids obtained by joining the points of each face to the center of the sphere.