Euclid — "Let it be granted that a circle may be described with any center and any radius."
Let it be granted that a circle may be described with any center and any radius.
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More from Euclid
"The greatest of the parts is called the antecedent, and the less the consequent."
"The angles in the same segment are equal to one another."
"To cut off from the greater of two given unequal straight lines a straight line equal to the less."
"What has been affirmed without proof can also be denied without proof."
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Understanding this quote
What it means
Any point in space can serve as the center of a circle, and any length can serve as its radius — no location is privileged, no scale forbidden. This postulate asserts that geometric space is uniform and unlimited: construction is always possible, regardless of where you are or how large or small you draw. It's a building-block assumption, not a proof — something granted so everything else can follow logically from it.
Relevance to Euclid
Euclid's entire career was devoted to building mathematics on unshakeable logical foundations. His Elements opens with definitions, postulates, and common notions — nothing assumed without declaration. This postulate mirrors his insistence that every geometric construction requires explicit permission. As a teacher in Alexandria, he reportedly told students seeking shortcuts that there is no royal road to geometry. Granting axioms precisely, then deriving everything rigorously, was his defining intellectual discipline.
The era
Around 300 BCE, Euclid worked in Alexandria under Ptolemy I, during the Hellenistic period when Greek culture spread across the former Persian Empire. Mathematics had been practical — Egyptian surveyors, Babylonian accountants — but Greek thinkers were transforming it into pure deductive science. Formalizing what counts as allowed in construction was radical: it separated geometry from physical tools and made it a system of pure logical consequence, independent of measurement error or physical imperfection.
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