Simon Bridge said:iirc: Weber's theory incorrectly predicts the speed of light and was shown to be inconsistent with the conservation of energy.[*]
Maxwel, on the other hand, managed to cover all of electromagnetism, including as a subset everything that Weber electrodynamics was useful for, and is useful in relativity as well as quantum mechanics.
Weber's extremely important and valuable work unifying electricity and magnetism is remembered by naming an unit after him.
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[*] I'll go look:
http://history.hyperjeff.net/electromagnetism
... Weber vanishes about 1870 when Hermann Ludwig Ferdinand von Helmholtz (1821–94) developes a theory of electricity and shows Weber's theories to be inconsistent with the conservation of energy.
J J O'Connor and E F Robertson said:A major topic which occupied Helmholtz after his appointment to Berlin was electrodynamics. He discussed with Weber the compatibility of Weber's electrodynamics with the principle of the conservation of energy. In fact the argument was heated and lasted throughout the 1870s. It was an argument which neither really won and the 1880s saw Maxwell's theory accepted. Helmholtz attempted to give a mechanical foundation to thermodynamics, and he also tried to derive Maxwell's electromagnetic field equations from the least action principle.
R Steven Turner writes in [1]:-
Helmholtz devoted his life to seeking the great unifying principles underlying nature. His career began with one such principle, that of energy, and concluded with another, that of least action. No less than the idealistic generation before him, he longed to understand the ultimate, subjective sources of knowledge. That longing found expression in his determination to understand the role of the sense organs, as mediators of experience, in the synthesis of knowledge.
To this continuity with the past Helmholtz and his generation brought two new elements, a profound distaste for metaphysics and an undeviating reliance on mathematics and mechanism. Helmholtz owed the scope and depth characteristic of his greatest work largely to the mathematical and experimental expertise which he brought to science. ... Helmholtz was the last great scholar whose work, in the tradition of Leibniz, embraced all the sciences, as well as philosophy and the fine arts.