  Dirac's quantization condition, $eg=n/2$ ($n \in \Bbb Z$), and Schwinger's
quantization condition, $eg=n$ ($n \in \Bbb Z$), for an electric charge $e$ and
a magnetic charge $g$ are derived by utilizing the Atiyah-Singer index theorem
in two dimensions. The massless Dirac equation on a sphere with a
magnetic-monopole background is solved in order to count the number of
zero-modes of the Dirac operator.
