Various Bayer Designation Arrangements
This "First to Rise in the East" method is done in a number of other instances, even for Castor and Pollux of Gemini. Although Pollux is brighter than Castor, the latter was assigned alpha because it rises in the east ahead of the former. Bayer may also have assigned the stars Castor and Pollux in terms of historical or mythological knowledge. Both historically and mythologically, Castor's name is almost always mentioned first (Castor and Pollux) whenever the twins are mentioned, and that may have compelled him to assign alpha (α) to Castor and beta (β) to Pollux.
Although the brightest star in Draco is Eltanin (Gamma Draconis), Thuban was assigned alpha (α) by Bayer because Thuban, in history, was once the north pole star, 4,000 years ago. Almost every star with a history of being the North Star, including Vega, Alderamin and Polaris, were designated as the alpha (α) of their parent constellations by Bayer.
Sometimes, indeed, there's no apparent order, as exemplified by the stars in Libra and Sagittarius, where Bayer assigned designations to stars at random.
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