Feynman diagrams will be looked at from a new point of view. 'Symmetries of Feynman Integrals' is an analytical method for calculating Feynman diagrams. It is based on exposing an underlying group structure of a given diagram which defines a set of partial differential equations in the parameter space of the diagram. Group orbits in the diagram’s parameter space are used to reduce the Feynman integral into a line integral. The vacuum seagull, a three-loop diagram, and the propagator seagull, a propagator-type diagram with two loops, will be used to demonstrate the method, and to obtain new results.