Newton's Fluxions

Newton's Fluxions

Using Integral Calculus and Binomial Expansions

Like Wallis before him, Newton decided to calculate by finding the area under the curve of a semicircle. Unlike Wallis, Newton finally had the tools to do this effectively: integral calculus and a broader binomial theorem. Newton began with a semicircle with a radius of 1/2 and centered at (1/2, 0):

First Newton decided to calculate the gray area under the curve AD. To do this he needed to be able to solve:

However, integrating the statement above was not simple; Newton had to find a way to deal with the complicated integrand. He was able to do this using his binomial theorem.

Before the time of Newton, Pascal and others had developed a method for dealing with the expansion of a binomial in the form , where c is an integer. The method was limited in that it relied upon the array of numbers known as Pascal's Triangle, and it wouldn't work for fractional or negative exponents. Newton developed a new binomial theorem which eliminated the shortcomings of the previous method. One version of Newton's famous binomial theorem is:

In more sophisticated terms, we can express this as:

Now let's get back to Newton's integral expression. Newton knew that the square root of a term can also be expressed as the one-half power of the term. We can take advantage of this fact in order to reëxpress the equation:

Then, using binomial theorem:

By distributing we get:

Finally, through using the methods of integration which Newton developed (and which he called "finding the Flowing Quantity from a fluxion") we learn that:

To finish our evaluation we just plug x = 1/4 into the equation:

Having gotten the gray area under the curve AD, we now need to find the area of triangle BCD (see diagram way above). We'll find the length of line segment BD by plugging into the equation for the semicircle:

Then we'll calculate the area of the triangle, using the age-old formula:

We now know the area for the sector which appears in gray in the diagram below.

We notice that the triangle whose area we have just calculated is a 30-60-90 right triangle. The angle of the sector is 60°, which means that the sector has 1/6 the area of the whole circle.

Finally, by solving for and inserting the area that we calculated for the sector, we get our approximation of :

We were able to calculate to three decimal places of accuracy using only the first five terms of the binomial expansion. Had we used nine terms of the expansion, we would have found seven correct places. Newton, himself, used twenty terms to accurately approximate to 16 places!

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