In this paper we present a theorem concerning the arc length on the Riemann sphere of the image of the unit circle under a rational function. But our larger purpose is to tell a story. We thought at first that the story began in 1962 with the Kreiss matrix theorem, the application that originally motivated us. However, our arc length question turns out to be more interesting than that. The story goes back to the famous "Buffon needle problem" of 1777.