As Sommerfeld said in his famous text "Spectral Lines and Atomic Constitution," on which a generation of physicists learned the subject, "In the fine structure constant e is the representative of the electron theory, h the appropriate representative of the quantum theory, c comes from relativity and characterizes it in contrast to classical theory.
Emilio Segrè, From X Rays To Quarks: Modern Physicists And Their Discoveries
constants-of-nature, fine-structure-constant, history-of-science, physics