I used Wikipedia to try to figure this out but I am not clear on the explanation. As for Newton's Law of Gravitation does not give the value for the Gravitational constant but it was measured by Cavendish.

I am wondering how the derivation was put together to arrive at the fine structure constant given that the units all cancel out or perhaps Sommerfeld calculated it's value.

Computed (as $\alpha=e^2/\hbar c$) in equation (12a), p. 51 of: Zur Quantentheorie der Spektrallinien, Ann. Physik (4) 51 (1916) 1–94.

(Earlier he had used $\alpha = (e^2/2\hbar c)^2$ in equation (16), p. 469 of Die Feinstruktur der Wasserstoff- und der Wasserstoff-ähnlichen Linien, Münch. Ber. 1915 (1916) 459–500. For discussion see e.g. M. Eckert (ed.) Die Bohr-Sommerfeldsche Atomtheorie, Springer (2013).)

  • Right from the horses mouth. very excellent. My German is a little rusty – Sedumjoy Sep 30 at 1:21
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    For English see his Atomic structure and spectral lines (1923, §VIII.2, p. 470), or Born (1927, §33). – Francois Ziegler Sep 30 at 1:57

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