The oscillatory component of the transverse magnetoresistance of bismuth has been measured as a function of magnetic field orientation at liquid helium temperatures. A derivative technique was employed. In addition to sets of periods, observed in the de Haas-van Alphen effect by Shoenberg and by Brandt and attributed by them, respectively, to the electron and light-hole Fermi surfaces, we observe a new set of isotropic short periods, P = 0.72X10~5 G" 1 . The squares of the electron Fermi momenta w 0 /cn 2 , w 0 /c 2 2 2 , W0K33 2 , 7W0K23 2 are, respectively, 0.189, 45.4, 0.918, and -4.54 mo milli-electron volts (meV). For the light holes, moKu 2 and W0K33 2 are 1.51 and 21.0 Wo meV. For the new heavy carriers, w 0 «: 2 = 3.18 Wo meV. These data are fitted to two possible three-carrier models of the Fermi surface, and to a four-carrier model. Significant deviations of the oscillations from periodicity in H~l are observed for the electron part of the Fermi surface for certain magnetic field orientations.