The methods recently developed by Schwinger for dealing with problems in quantum electrodynamics have been applied to an investigation of the electromagnetic properties of nucleons in interaction with pseudoscalar mesons. The calculated values of the magnetic moments of the neutron and proton are in agreement with results previously obtained by Case, Luttinger, and Slotnick and Heitler using different calculational procedures. Our result for the neutron-electron interaction is identical with that obtained by Case. Included in this paper are the anomalous proton-electron interaction and a qualitative discussion of the scattering of high energy nucleons. The large coupling constants necessary to fit any of the experimental data show that the results obtained by a second order perturbation calculation are not quantitatively reliable.