On 08/02/2013 06:49 AM, Richard Tobin wrote: > In article <firstname.lastname@example.org>, > <email@example.com> wrote: > >> But one 1/phi is not a rational ratio, but the 1/6 hexagon (center to >> vertex/sum of sides) is and so is all polygons derived from multiplying >> the vertices. > > You were claiming that pi is rational. It is not. > > -- Richard >
This jogged my memory about Lagrange and irrational numbers. Lambert was the first to prove the irrationality of pi: in 1761,
Lagrange showed that if x > 0 is rational, then sin(x) is irrational.
Ref.: (from sci.math in 2011) ====================================================================== In Jean-Guillaume Garnier's "Analyse algebrique, faisant suite a la premiere section de l'algebre", 1814,
pages 538 and 539, he mentions Lagrange and C. Haros. He writes that Lagrange showed that, if x > 0 is a rational number, say x = p/q with gcd(p, q) = 1, then sin(x) is irrational.