From: utzoo!decvax!harpo!floyd!vax135!ariel!orion!lime!burdvax!psuvax!sibley
Newsgroups: net.math
Title: Re: Squares
Article-I.D.: psuvax.1160
Posted: Fri Jan 28 14:50:15 1983
Received: Tue Feb  1 10:49:40 1983


The article uiucdcs.1389 suggests a way to determine the length of the
diagonal of a square.  It doesn't work, but the author wants to know why.

First, it is true that it doesn't work.  In the example given, the sum of
all the lengths of the little vertical pieces is always 1, the original
side length, and the same happens horizontally.  That's why you get 2.

This explains why the integral formula for arc length of a curve is so
complicated.  That is, if the suggested procedure worked, one could
calculate arc length by integrating ( y' + 1 )dx instead of the usual 
(((y')**2 + 1 )**.5)dx.  Note that the first (wrong) integral is actually
easy to evaluate -- it is
	y(b) - y(a) + b - a
if the integral is taken from a to b.  The second (correct) one does not
simplify.

Dave Sibley
Department of Mathematics
Penn State University
psuvax!sibley