Friday, November 12, 2010

Newton's approximation for pi

Borrowing shamelessly from Journey Through Genius, we'll finish the example of Newton's version of the binomial expansion from here.

Start with a semicircle as shown in the diagram below. The equation of the circle is

(x - 1/2)2 + (y - 0)2 = (1/2)2
x2 - x + 1/4 + y2 = 1/4

Solve for y:

y = √(x - x2)
= √x √(1-x)



The positive square root is the equation of the semicircle, where √(1-x) can be replaced by its binomial expansion:

1 - (1/2)x - (1/8)x2 - (1/16)x3 - (5/128)x4 ..

and multiplied by x1/2 to give:

x1/2 - (1/2)x3/2 - (1/8)x5/2 - (1/16)x7/2 - (5/128)x9/2 ..

Now let B be the point (1/4,0) as shown.

The area of the arc of the circle ACD in the diagram comprises two areas: the triangle BCD plus the area ABD. Newton used fluxions to calculate the area of ABD. In modern terminology, we integrate the equation of the semicircle given by the binomial expansion:

(2/3)x3/2 - (1/2)(2/5)x5/2 - (1/8)(2/7)x7/2 - (1/16)(2/9)x9/2 - (5/128)(2/11)x11/2 ..

evaluated for between x = 0 and x = 1/4. Now the reason for picking these particular coordinates is revealed:

(1/4)3/2 = (√(1/4))3 = 1/8
(1/4)5/2 = (√(1/4))5 = 1/32
..

So the series above reduces to:

1/12 - 1/160 - 1/3584 - 1/36846 .. = 0.07677

(more precision shown in Dunham's book, achieved easily by Newton). Using the Pythagorean theorem instead we get that

BD = √((1/2)2 - (1/4)2) = √(3/16) = √3/4

And the area of the triangle BCD is

(1/2) √3/4 (1/4) = √3/32

Since the base of BCD is 1/4 and the radius is 1/2, the cosine of the angle ACD (BCD) is 0.5 and ACD equals 60 degrees, meaning that the total area of ACD is 1/6 of the circle:

π/24

Using the value obtained from the series above:

π = 24(0.07677 + √3/32)
= 3.1415..

A contemporary recorded this account of Newton's Lucasian lectures: "so few went to hear him, & fewer yet understood him, that oftimes he did in a manner, for want of Hearers, read to ye Walls." The commentator added that Newton's lectures would last for half an hour except when there was no one at all in the audience, in that case he would stay only 15 minutes.

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