We describe a method for measuring the shape of a toroidal magnetic field (such as in a stellarator) to demonstrate the existence and nature of magnetic surfaces. We detect the successive circuits of a pulse of low energy, phase stabilized electrons injected parallel to the field and stored for many thousands of circuits. We find that the cumulative effect of various field perturbations alters the parameters of the surfaces significantly, but does not destroy them. In certain special cases, there is excellent agreement between the measured and computed field shapes for our specific device. These results and methods are of immediate application to a broad class of magnetic (or other area preserving) mappings. The same experimental technique can also be used for the long term storage of low energy ions.