Squaring the Circle

Date: 3/16/96 at 21:44:27
From: Deborah A. Lindsay 
Subject: math question

Dr. Math,

Where did the phrase "squaring the circle" come from?  
We found it in literature and wonder about its origins and 
what it means.

Thank you.

(United States House of Representatives Page School)

Date: 3/17/96 at 1:22:24
From: Doctor Jodi
Subject: Re: math question

Hello! Thanks for your question.  

The Greeks were great at posing mathematical questions that 
no one could solve.  One of these questions was:

Using only a straight edge and compass, construct a square 
with the same area as a circle. The Greeks tried to solve this 
by construction as early as 200 BC.

From this problem comes the phrase you ask about, 
"squaring the circle".

Now, we know that the constant that relates the area of a circle 
and its perimeter to the radius, pi (usally represented by the 
Greek letter of the same name), is a transcendental number.  

Transcendental, you say, that's an interesting name for a number, 
but what does it mean?

Well, a transcendental number is one that can't be expressed as 
a ratio of two integers AND it can't be expressed as a terminating 
decimal, like 1.09 or a non-terminating decimal with a pattern: 
i.e. 0.3333333, etc. or 1.297297297297, etc.

In fact, we have calculated a few MILLION digits for pi and we 
still find it to be transcendental.  But more importantly, people 
have proven that pi is transcendental. I think that the first proof 
of this was by Lindemann in 182 (described in Courant and 
Robbins' _What is Mathematics_).

So that ancient problem which so many people have struggled 
with, squaring the circle, is not possible with just compass and 
straight edge.

Hope that this answers your question...  

I'm really curious about what the Page School is (a school for 
would-be pages? for the children of House members?)

-Doctor Jodi,  The Math Forum