What’s the origin of the phrase ‘Fair and square’?
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This square has nothing to do with geometry.
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In the 16th century ‘square’ meant ‘fair and honest’ so ‘fair and square’ is tautological.
George Puttenham used that meaning of square in The arte of English poesie, 1589:
“[Aristotle] termeth a constant minded man - a square man.”
Francis Bacon’s essay Of Prophecies, 1604 is the first known use of ‘fair and square’:
“Faire, and square. The gamester calls fooles holy-day.”
See also: the last words of Sir Francis Bacon.