Introduction.

There is a close and beautiful connection between the transformation theory for elliptic integrals and the very rapid approximation of

. This connection was first made explicit by Ramanujan in his 1914 paper ``Modular Equations and Approximations to

''[26]. We might emphasize that Algorithms 1 and 2 are not to be found in Ramanujan's work, indeed no recursive approximation of

is considered, but as we shall see they are intimately related to his analysis. Three central examples are: