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A number multiplied by itself, or raised to the second power. The square of three is nine; the square of nine is eighty-one.
To make things right, just, proper, etc: He could never square himself with the police after that (1859+)
[the sense ''conventional person, etc,'' is said to come fr a jazz musician's and standard conductor's hand gesture that beats out regular and unsyncopated four-beat rhythm, the hand doing so describing a square figure in the air]