The long-awaited textbook on ECE theory (paper 438) has been expanded by electrodynamics. The first chapters contain the basic mathematics underlying ECE theory and an introduction to Cartan geometry. The part on electrodynamics introduces the physical interpretation of Cartan geometry, extending the scope from former Maxwell-Heaviside theory to a theory of general relativity. The field equations and new aspects like antisymmetry laws are described. The final chapter with ECE2 theory, a further development, and longitudinal waves is still missing.