The rise of dimensional analysis in the latter part of the nineteenth century occurred largely in the context of electromagnetism. It soon appeared that the subject, albeit seemingly straightforward, was in fact wrought with difficulties. These revealed deep conceptual issues regarding the character of physical quantities. Usually, whether or not these problems actually constituted inconsistencies was itself a matter of debate. In one instance, however, regarding the electrostatic dimensions of the magnetic pole, all protagonists agreed that the matter required attention. (...) A controversy ensued in 1882. Its resolution partly relied on the realization that it arose from differences between the scientific cultures prevalent on the Continent versus in Great Britain. These cultural differences concerned the possible relevance of the medium in which interactions involving magnetic poles take place, as well as the understanding of permeability in Ampère’s model of magnetism. The controversy around the electrostatic dimensions of the magnetic pole entailed crucial issues that were soon to play just as central a part in a wider debate about the dimensions of electromagnetic quantities in different systems of units. Why the latter topic was never raised during the 1882 controversy provides insights into the early understanding of dimensional analysis. (shrink)