Origins
Precision is a cutting art. The word precise comes from Latin praecīsus, the past participle of praecīdere — 'to cut off in front' — from prae- ('before') and caedere ('to cut'). A precise statement is one where everything unnecessary has been cut away, leaving only what matters.
The Latin verb caedere ('to cut, to strike, to fell') generated an entire vocabulary of intellectual sharpness. Concise means 'cut together' — compressed. Decide means 'cut off' — to eliminate alternatives. Incise means 'cut into' — to engrave. Excise means 'cut out' — to remove. Even scissors descends from a related Latin form, cisōria ('cutting instruments').
The connection between cutting and thinking runs deeper than metaphor. Roman rhetoricians valued brevitas — the ability to cut a speech to its essential points. A speaker who was praecīsus was admirably brief, not wastefully exact. The modern sense of 'exact, accurate' developed in the 16th century, when the word shifted from how much you removed to how accurate what remained was.
Later History
French preserved both meanings. A précis is a summary — a text cut to its core. The adjective précis means 'exact'. English borrowed the summary sense as précis and the accuracy sense as precise, splitting one French word into two English uses.
The earliest English citations show precise used to describe people, not measurements. A precise person was fastidious, fussy, overly strict. The Puritans were called 'the precise ones' — a criticism, not a compliment. The neutral, technical sense came later.