The global (or homology) version of Cauchy’s theorem was given an elementary proof by John Dixon. I believe this is mentioned in Rudin's Real and Complex Analysis. A proof is available online at http://www.math.uiuc.edu/~r-ash/CV/CV3.pdf. This states "The elementary proof to be presented below is due to John Dixon, and appeared in Proc. Amer. Math. Soc. 29 (1971), pp. 625-626, but the theorem as stated is originally due to E.Artin."