**Reply to Harry, Diminishing Circles Argument**

**Jesse L. Yoder**

** **

**Subject:** Reply to Harry, Diminishing Circles
Argument

**Author:** Jesse Yoder jesse@flowresearch.com

**Organization:** Flow Research

**Date:** 31 Oct 1997 11:52:02 ñ0500

Hi Harry ñ

You wrote the following:

"I
enjoyed reading this and thinking about it. You have started with a semicircle,
divided it into two semicircles, divided each of those into two semicircles,
etc., while the total length remained the same. You could have started with an
isosceles triangle instead of a semicircle and performed similar operations.
Your argument would then say that the diameter of the circle is two times the
length of a side of the original triangle. This argument would then assert that
the diameter of a triangle is any number greater than half the circumference.

The
problem here is the following. If you have a sequence of functions f(n) on an
interval (whose length is the length of the diameter) which converge to another
function f on the interval (in this case the zero function), convergence either
pointwise or uniform, it does not mean that the lengths of the graphs of the
f(n)'s converge to the length of the graph of f. To see why this is true one
need only to look at the formula for arc length and consider your most
interesting example."

Thanks for replying to my
diminishing circles argument. I'm not sure, however, that I completely
understand your parallel example of the isosceles triangle. First of all, do
you mean that I should draw an isosceles triangle inside a semicircle? If I
draw one isosceles triangle inside a semicircle, with the tip of the triangle
intersecting the circle circumference, the distance "around" the
triangle is not equal to the distance around half circumference. Of course, if
we are to believe calculus, if we continue to draw isosceles triangles one on
top of the other, with the tips touching the circumference of the circle, the
limit of these triangles approaches the circle circumference. But it is precisely
this way of determining the area of a circle that I object to.

At the end of your first
paragraph, do you mean "diameter of any circle" instead of
"diameter of any triangle"?

Possibly your point about
functions and their lengths comes down to this: The limit of any function as it
approaches infinity does not necessarily have the same properties as the
function before the limit. If this is your argument, then if the diameter of
the circle is the limit of the diminishing circles, the diameter does not necessarily
share the same length as the diminishing circles before they reach the
diameter. While the diminishing circles have the same "distance
around" as the circumference/2 until they reach their limit, at the limit
they no longer have this property.

In response to this, please note
that I do not rely on the idea that the function "goes to infinity";
instead, the "limiting case" of the circles is when the circles
become the points on the number line. Isn't there a problem that the circle has
area while a point does not? In my geometry, these points have area, and the
unit of measurement that is specified when the circle is created (or the
"line" is drawn) determines their area. When the circles become as
small as points on the number line (as specified for each line), this is their
limit and they actually become identical to the points on the line.

As I said before, however, it is
vital to my argument that a line be viewed as "bumpy"; as consisting
of a set of points packed up against each other. If a line is instead viewed as
the path of one of these points in motion, giving you the effect of a straight
line, then my argument will not work. This is because traveling along the
diameter will no longer be traveling up and down these points, but instead will
be along a "straight line" that intersects these points at the tip of
the circumference of the circles that lie on the line.

Thanks again for your challenging
comments. Possibly I have strayed from the point of your example, but I hope
that you will explain it further. Also, please clarify or elaborate on your
comment about the formula for arc length.

Best wishes,

Jesse

To view original text, go to:

http://forum.swarthmore.edu/epigone/geometry-research/khixskarban

`Jesse L. Yoder`

`Flow Research`

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