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1907, "bald person," from egg (n.) + head (n.). Sense of "intellectual" is attested from 1918, among Chicago newspapermen; popularized by U.S. syndicated columnist Stewart Alsop in 1952 in reference to Adlai Stevenson's presidential campaign.
Adlai Stevenson once told what it was like to be the rare intellectual in politics. "Via ovicapitum dura est," he said, the way of the egghead is hard. [New York Times, Oct. 28, 1982]
[second sense presumably fr the putative high, domed, egg-shaped heads of such persons; the term was used in a letter of Carl Sandburg about 1918]