  The development of relational electromagnetism after Gauss appears to stop
around 1870. Maxwell recognised relational electromagnetism as mathematically
equivalent to his own formulae and called for an explanation of why so
different conceptions have such a large part in common. We reconstruct
relational electromagnetism guided by the No Arbitrariness Principle. Lorenz'
idea of electromagnetic waves, together with the "least action principle"
proposed by Lorentz are enough to derive Maxwell's equations, the continuity
equation and the Lorentz' force. We show that there must be two more symmetries
in electromagnetism: a descriptive one expressing source/detector relations,
and another relating perceptions of the same source by detectors moving with
different (constant) relative velocities. The Poincar{\'e} group relates
perceived fields by different receivers and Lorentz boosts elate
source/detector perceptions. We answer Maxwell's philosophical question showing
how similar theories can be abduced using different inferred entities. Each
form of abduction implies an interpretation and a facilitation of the
theoretical construction.
