July 2014 – Hi Math

Where does the symbol π come from?

In 1652, William Oughtred used π to refer to the periphery of a circle (in his expression, the ratio of circumference-to-diameter of a circle is π ⁄ δ, the latter referring to diameter).

In 1665, Jonh Wallis used a Hebrew letter (mem) to equal the ratio of one-quarter of circumference to diameter of a circle. (This letter plays the role as of “M” in Latin alphabets, but look how close its shape resembles a quarter of a circle, as well as the Greek letter pi !)

In 1705, William Johns used π to represent the ratio of circumference-to-diameter of a circle (believed to be first use with exactly same meaning as in today) .

From 1736, Leonard Euler, both famous and a prolific writer in mathematics works, spread the use of π in his publications.

Counting from the first relevant use, the symbol π has already had a history of more than 360 years!

The picture below actually shows the proof.

The goal is to show the red line (which is supposed to bisect circular area) is longer than the diameter AB’.
We see it from:

Length of red curve AB is greater than: AE + EB = AE + EB’

which is certainly longer than AB.

Just one more minute. Let’s review the construction-proof process.

Connect the two endpoints A, B of the curve by a line segment. Draw diameter CD // AB.

Take O (the midpoint of CD), then passing A and O we get another diameter AB’.

The two arguments needed for completing proof are:

(1) Red line must have at least one intersection with diameter CD; (think like this: if the red line resides completely at only one side, then there is no point that it can evenly divide the circular area) Suppose the intersection is at point E;

(2) B and B’ are symmetric to the diameter CD therefore EB = EB’ (think on why? using the property of circles and parallel lines)

Now the rest of the proof is straight forward.