In a very obvious way that shows no attempt to hide something, often something wrong or offensive.
From 'blatant,' which appeared in English around 1590, possibly from Latin 'blatire' (to bleat), the idea being loudly obvious like a bleating sheep. The suffix '-ly' makes it an adverb.
Blatant shows up suddenly in English in the 1590s and no one's sure why—it might be Edmund Spenser inventing it!—but once it appears, writers immediately understood it meant 'obviously wrong' and it stuck forever.
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