In 1925 Samuel Goudsmit and George Uhlenbeck were studying the atomic spectra of alkali atoms such as Li and Na, they showed that in addition to its orbital motion, an electron must be regarded as spinning about some axis. The story is told by Goudsmit in a lecture nearly fifty years later:

"When the day came I had to tell Uhlenbeck about the Pauli principle - of course using my own quantum numbers - then he said to me "But don't you see what this implies? It means that there is a fourth degree of freedom for the electron. It means that the electron has spin, that it rotates." ... Then I said : "That fits precisely in our hydrogen scheme ... and if now allows the electron to be magnetic with the appropriate magnetic moment, then one can understand all those complicated Zeeman-effects. They come out naturally, as well as the Landé formulae and everything, it works beautifully"
Samuel Goudsmit in April 1971

G.E.Uhlenbeck and S.Goudsmit, Naturwissenschaften, 1925, 47, 953. See : Lecture by Samuel Goudsmit the the Dutch Physical Society in April 1971, and celebrating the Golden Jubilee of the Society. Translated by J.H. van der Waals, and printed in "Foundations of Modern EPR" by G.R.Eaton, S.S.Eaton and K.M.Salikov

"The discovery of the electron spin" 1971 Dutch Physical Society Lecture by S.A.Goudsmit,