Gabriel's Horn. Was: capacitor or condenser ?

["Donald E. Simanek" <dsimanek@eagle.lhup.edu>]

Thank you, Brian, for giving us the name by which this class of puzzlers
is called: Gabriel's Horn. Now we have a handle for seeking out
discussions of it in the literature, and extensions of it. And thanks to
Ludwik for stating the apparent paradox in it's most interesting form, the
amount of paint needed and the liquid needed to fill it. Now extend the
problem one more step. Why is the partition needed to make the startling
conclusion? Isn't the surface of revolution of the curve a vessel? and
isn't it's surface area simply calculable by the Theorem of Pappus and
Guildinus? Or other appropriate method.

	-- Donald

On Sat, 5 Apr 1997, brian whatcott wrote:

> At 12:13 PM 3/31/97 EDT, LUDWIK KOWALSKI wrote:
> > I was sharing our "meridian triangles paradox" with a friend this morning
> > and became aware of another one. He said it is a well known "puzzle".
> >
> > Plot the y=1/x curve and try to calculate the area below, by integration.
> > The answer is infinity...
> > ... revolve the
> > curve about the x axis and calculate the volume. It is finite....
> >                                                   Ludwik Kowalski
>
> Line length, area and enclosed volume of 
> y = 1/x for x>1
>  is called the "Gabriel's Horn" problem, sometimes given as a
> proposition to be proven.
>
> Regards
> brian whatcott <inet@intellisys.net>
> Altus OK

From: LUDWIK KOWALSKI <kowalskil@alpha.montclair.edu>

But I am really responding to Brian who wrote:

> Line length, area and enclosed volume of y = 1/x for x>1 is called the 
> "Gabriel's Horn" problem, sometimes given as a proposition to be proven.

This horn has a paradoxial property. Divide it into two parts by inserting 
a single plane partition which contains the x axis. You have a vessel whose 
one wall can take as much paint as you wish (an infinite area, when x --> 
infinity) but whose volume capacity is always finite.

                               Ludwik Kowalski