The following calculation of is due to William Brouncker (1620-1684)[22, page 54].

Let
denote the
regular partition of the interval into equal subintervals.
Let

We can construct from by adjoining a box of width to the top of each box that occurs in the definition of (see figures a) and b)).

From figure a) we see that

From figure b) we see that

In general we will find that

Now

i.e.

Thus

We can think of as being given by the ``infinite sum"

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
not very
useful for calculating: it takes too many terms in the sum to get much
accuracy.
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
formula
as giving a ``closed form" for the sum of the infinite series
.