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Origin documented
“Won't budge an inch”
What it means
Absolutely immovable.
Where it comes from
The inch works because it is the smallest unit in common speech. Refusing to move even that far is a claim about principle rather than distance.
How sure are we?
The construction appears in Shakespeare and has been in continuous use since. The inch itself was fixed in 1324 as three barleycorns end to end.