Archimedes — "The value of pi is more than 3 10/71 and less than 3 1/7."
The value of pi is more than 3 10/71 and less than 3 1/7.
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Verification
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Understanding this quote
What it means
Pi cannot be written as a simple fraction, so rather than guessing, this statement proves pi lives inside a tight, verifiable range — above 3.1408 and below 3.1429. It replaces speculation with geometric certainty. The approach is what modern scientists call error bounds: not claiming perfect knowledge, but establishing proven limits. Rigorous, honest, and more useful than a false exact answer.
Relevance to Archimedes
Archimedes derived this bound by inscribing and circumscribing 96-sided polygons around a circle and computing their perimeters entirely by hand — no decimal notation, no calculators. It reflects his defining character: methodical proof over intuition. The same discipline appears across his work on sphere volumes, lever mechanics, and buoyancy. He never stated what he could not prove, only what geometry compelled him to accept.
The era
Archimedes lived in Syracuse around 287–212 BC, when Greek mathematics was maturing under Euclid's framework of formal proof. Earlier civilizations — Babylonians and Egyptians — used rough pi estimates near 3.1 with no justification. Greek culture demanded logical rigor, yet lacked decimal notation or algebra, making tight arithmetic bounds extraordinarily laborious. Archimedes's result, accurate to two decimal places, stood as the most precise Western approximation for roughly fifteen hundred years.
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