Wallis' Formula

Wallis' Formula

The Area of the Quadrant of a Circle Before Integral Calculus

John Wallis (1616-1703) discovered a method for calculating the value of by finding the area under the quadrant of a circle. Some evidence suggests that a similar method was also used in Japan in the late 17th Century. A quadrant of a unit circle has the area of /4. By finding that area, one can find the value of . Geometrically, Wallis was trying to find the grey area in the diagram below.

In modern calculus notation what Wallis was working with was:

In 1671, Newton was able to do this calculation. However, when Wallis was trying to solve this problem, integral calculus had not yet been invented and binomial theorem had not yet been worked out for non-integers. So Wallis, through a long series of interpolations and inductions derived what is now known as Wallis' Formula:

Wallis' Formula is remarkable because it is the first infinite series for that did not involve irrational numbers, like square roots. Carrying this series out to the 60th term (60 in the numerator, 61 in the denominator), we get an approximation of = 3.11595. Clearly, this isn't too accurate. Nevertheless, the calculations are easy enough that the series can be expanded more quickly than more complicated ones. The more terms we carry the equation out to, the more precise our approximation will be.

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