The discussion of the historical background of the 19th century electromagnetic theory has shown that from the standpoint of modern scientific method, the Hertz experiments on propagation of electromagnetic interactions cannot be considered as conclusive at many points as it is generally implied. It has been found that alternative Helmholtz's electrodynamics did not contradict Hertz's experimental observations. Mathematical analysis of the conventional electromagnetic theory showed that numerous ambiguities are related to the treatment of time behaviours. Those difficulties turn out to be cleared up by distinguishing between implicit and explicit time dependencies. It provides self-consistency for mathematical description of electromagnetic theory by advocating the explicit use of full time derivatives in the mathematical formulation of Maxwell's equations. This approach covers conventional electromagnetic theory based on partial time derivatives. The covering theory is showed to possess all necessary relativistic invariance properties for inertial frames of references. The idea of non-local interactions is enclosed into the framework of Helmholtzian electromagnetic theory as unambiguous mathematical feature. In this work we make a point that Helmholtz's foundations and modern Helmholtz-type electrodynamics recently developed by the authors and reviewed here promise, in general, an altogether more logical solution to self-consistent classical electrodynamics and its reconciliation with quantum mechanics.