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I do not think the transistor effect was found by experimentalists..
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http://en.wikipedia.org/wiki/Julius_Edgar_Lilienfeld
Crooks tubes --> vacuum tubes Tesla, Eddington had their own --> Lillienfield electrolyte capacitors --> Field Effect
Field emission existed before it was explained by Quantum Tunneling no?
To suggest it was a giant leap by a smart man with a biro is a bit rich.
Since when has mother nature given a stuff about man's mathematics?
Ofcourse Tesla wrote papers to describe what he had experimented with. He was not a desk jockey developing the maths of AC circuits with a graph paper and a pad... then to telegraph it to a manufacturer.
He built it himself, he wound his own coils, he was an experimenter.
Documenting his experiments, in patents, is just the record of his empirical science. Did he have ideas of where to take his research, well ofcourse... was he a mathamatical theorist? No! he hated them!
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TESLA: Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.
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Teslas empirical physics is FAR FAR different from string theory, multiverses, dark energy, dark matter, gravity waves which are abstract mathematical thought experiments...