Newton's Series for Pi

Equation (2.3.8) requires many terms for a decent approximation to . Newton made his calculations much easier by applying his binomial series to the shaded area from x = 0 to x = 1/4 below the graph as shown in the figure above.
On the one hand, this area can be approximated by a series:

On the other hand, this area is one-sixth of a circle of radius 1/2 minus a right triangle whose base is 1/4 and whose hypotenuse is 1/2:

The square root of 3 can be approximated using the binomial series:

Putting these together, we see that