  This paper expounds very innovative results achieved between the mid-14th
century and the beginning of the 16th century by Indian astronomers belonging
to the so-called "M\=adhava school". These results were in keeping with
researches in trigonometry: they concern the calculation of the eight of the
circumference of a circle. They not only expose an analog of the series
expansion of arctan(1) usually known as the "Leibniz series", but also other
analogs of series expansions, the convergence of which is much faster. These
series expansions are derived from evaluations of the rests of the partial sums
of the primordial series, by means of some convergents of generalized continued
fractions. A justification of these results in modern terms is provided, which
aims at restoring their full mathematical interest.
