The magnetic field on any closed surface can be separated into the component produced by currents in the volume enclosed by the surface and the component produced by currents external to the surface. This decomposition was first discovered by Gauss in 1839, and “Gauss' Algorithm” was used to show that the terrestrial field is sourced from inside the Earth. This magnetic source-decomposition can be used in magnetic confinement fusion to distinguish plasma fields from the fields of external sources using standard magnetic measurements, when no conductors reside in the region between the plasma and the magnetic sensors (otherwise, the internal source is not the plasma alone). Here, Gauss' Algorithm is extended from spherical geometry to cylindrical geometry to decompose fields in the cylindrical tokamak. Demonstrations of the decomposition on DIII-D discharges include the measurement of a rotating tearing mode separate from the induced wall eddy currents and the recovery of a vacuum measurement in the presence of an externally applied 3D field. Anticipated axisymmetric and non-axisymmetric applications, and the requisite diagnostics, are discussed. It is concluded that the ITER magnetic diagnostic design is well suited for many important axisymmetric and non-axisymmetric applications. Future work will extend this decomposition to include fields from halo currents and to generalize it to arbitrary toroidal geometries.