From: hale@tulane.edu (William Hale)
Subject: Re: Definition of pi
Date: Wed, 05 Dec 2001 20:41:49 -0600
Newsgroups: sci.math
Summary: How to define lengths of non-straight curves? (Archimedes solution)

In article <4OzP7.632$tg4.7225@vixen.cso.uiuc.edu>, prussing@uiuc.edu
(John E. Prussing) wrote:

> In <051220011159350410%hunsinger@mac.com> Ron Hunsinger
<hunsinger@mac.com> writes:
[cut]
> >But that requires the introduction of a preferred unit of distance,
> >something which I find disquieting. If you must define mathematical
> >constants geometrically, I much prefer that they be defined as ratios.
> 
> >As I recall, Euclid proves that the circumference of a circle lies
> >between the perimeters of the inscribed regular n-gon and the
> >circumscribed regular n-gon, and that the ratios of these perimeters to
> >the diameter of the circle can be made as close together as needed by
> >making n large enough. It follows that the ratio of the circumference
> >to the diameter of any circle must be the unique number lying between
> >these sequences. All done without needing to define a preferred "unit
> >length".
> 
> >-Ron Hunsinger
> 
> But it's incorrect to claim that as one curve approaches another in the
> limit, their arc lengths do also.
> 
> If you take a path in a unit square starting from the lower left corner
> and terminating at the upper right corner, with the stipulation that it
> travel in equal-length segments alternating between going up and going
> to the right (a staircase function), the total length of this path is
> 2, independent of n, the number of steps. But as you let n approach
> infinity the path approaches the diagonal, which has length sqrt(2).
> If you claim that the path length on the staircase function approaches
> the path length on the diagonal in the limit, then 2 = sqrt(2), from
> which it follows that 2 = 1, which we can all agree is incorrect.
> 
> Maybe the fact that neither the inscribed nor the circumscribed
> regular n-gons cross the circle makes the Euclid proof valid.

Archimedes handled this problem by introducing the following axiom:

  "Of other lines in a plane and having the same extremities,
   [any two] such are unequal whenever both are concave
   in the same direction and one of them is either wholly
   included between the other and the straight line
   which has the same extremities with it,
   or is partly included by, and is partly common with, the other;
   and that [line] which is included in the lesser [of the two]."

Your path of lenght 2 is not concave, so it cannot be used to
approximate the lenght of the diagonal.

I became familiar with this axiom when I taught a course on the
history of mathematics. I am not sure if Euclid was aware that
he was using something like this.

I didn't recall the exact statement of Archimedes axiom. I did
a google search on "concave curve" since that is what I remembered.
I found the above quote on a web page that talks about the topic
of this thread. The web page is at:

    http://www.ihes.fr/~ilan/pi-exists.html


--
Bill Hale
==============================================================================

From: hale@tulane.edu (William Hale)
Subject: Re: Definition of pi
Date: Wed, 05 Dec 2001 20:51:52 -0600
Newsgroups: sci.math

In article <hale-0512012041490001@dialup-4-83.tulane.edu>, hale@tulane.edu
(William Hale) wrote:
> Archimedes handled this problem by introducing the following axiom:
> 
>   "Of other lines in a plane and having the same extremities,
>    [any two] such are unequal whenever both are concave
>    in the same direction and one of them is either wholly
>    included between the other and the straight line
>    which has the same extremities with it,
>    or is partly included by, and is partly common with, the other;
>    and that [line] which is included in the lesser [of the two]."

Sorry, I guess I should have proofread the above. That last "in"
should be an "is".

Thus, the axiom of Archimedes is:

   "Of other lines in a plane and having the same extremities,
    [any two] such are unequal whenever both are concave
    in the same direction and one of them is either wholly
    included between the other and the straight line
    which has the same extremities with it,
    or is partly included by, and is partly common with, the other;
    and that [line] which is included is the lesser [of the two]."

Of course, "line" here means "curve".

Applied to the circle, note that the circle, circumscribed polygons,
and inscribed polygons are all concave "lines" that meet the
above criteria (at least the upper halves).

--
Bill Hale