  Analytic methods for Abelian projection are developed. A number of results
are obtained related to string tension measurements. It is proven that even
without gauge fixing, abelian projection yields string tensions of the
underlying non-Abelian theory. Strong arguments are given for similar results
in the case where gauge fixing is employed. The methods used emphasize that the
projected theory is derived from the underlying non-Abelian theory rather than
vice versa. In general, the choice of subgroup used for projection is not very
important, and need not be Abelian. While gauge fixing is shown to be in
principle unnecessary for the success of Abelian projection, it is
computationally advantageous for the same reasons that improved operators,
e.g., the use of fat links, are advantageous in Wilson loop measurements. Two
other issues, Casimir scaling and the conflict between projection and critical
universality, are also discussed.
