Gregory, Leibniz, and Machin

Gregory, Leibniz, and Machin

Using the Arctangent Infinite Series

Two mathematicians, James Gregory (1638-1675) and Gottfried Wilhelm Leibniz (1646-1716), each working independantly of one another, discovered an infinite series for arctan x in the 1670's. This arctangent series led to the first ever found infinite series for . Here is the arctangent series that they discovered:

For clarity, it can be rewritten as:

By expanding this series to an arbitrary number of terms we can approximate any arctangent to arbitrary precision. Gregory and Leibniz then found an equation for in terms of the arctangent. Here is a derivation of that equation:

Using the diagram of a unit semicircle above, it can be shown that:

Let's define tan in terms of sin and cos using the values we just calculated:

Since tan /4 = 1, then arctan 1 = /4. Going back to the arctangent formula, let's replace x with 1 and then simplify:

This famous series, often called the Gregory/Leibniz series, was the first infite series ever discovered for . However, the problem with the series is that its convergence is so slow that it is practically useless. 300 terms are insufficient to obtain even two decimal places of accuracy. In order to achieve 100 accurate decimal places, one would have to go through 100 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 iterations of the series!

Machin's Variation

In 1706, John Machin introduced a variation on the Gregory/Leibniz series with a significantly increased rate of convergence, which made it a much more practical method of calculation. It's so effective that Machin's formula is one that is still used today by computer programmers to calculate the digits of . I wrote a program in C++ which is based on Machin's forumla, and I used it to calculate the first 8192 digits of . Here is the derivation of Machin's formula. We start by defining by:

Then we use the double-angle forumula for tan to find:

Using the tan double-angle formula again we learn:

We note that this number, 120/119, is only 1/119 off from 1, which has an arctan of /4. Machin decided to calculate this this difference (120/119 - 1) in terms of the angles:

Therefore:

Now, we can substitute for it's arctangent value:

Finally, by solving for /4 we get:

Using his own formula, Machin was able to calculate 100 places of in 1706.

Since Machin's time many many other arctangent formulae for pi have been discovered. The formulae are, like Machin's formula, just the sum or difference of a few arctangent values.

Copyright 1996-1998 Paul A. Gusmorino 3rd. All Rights Reserved.