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