What’s the origin of the phrase ‘Fair and square’?
This square has nothing to do with geometry.
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.