The following calculation of is due to William Brouncker (1620-1684)[22, page 54].
regular partition of the interval into equal subintervals.
Equation (5.82) is sometimes called Mercator's expansion for , after Nicolaus Mercator, who found the result sometime near 1667 by an entirely different method.
Brouncker's calculation was published in 1668, but was done about ten years earlier [22, pages 56-56].
Brouncker's formula above is an elegant result, but it is
useful for calculating: it takes too many terms in the sum to get much
Today, when a logarithm can be found by pressing a button on a calculator,
we tend to think of ``" as being a known number, and of Brouncker's
as giving a ``closed form" for the sum of the infinite series