A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.
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"A point is that which has no part."
"An obtuse angle is an angle greater than a right angle."
"If a straight line touch a circle, and from the point of contact there be drawn across in the circle a straight line cutting the circle, the angles which it makes with the tangent will be equal to the…"
"That all right angles are equal to one another."
"A straight line is that which lies evenly between its extreme points."
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A circle's diameter is any straight line passing through its center, with both endpoints touching the boundary. Critically, this line divides the circle into two perfectly equal halves. The statement does double duty: it defines a term precisely and asserts a geometric truth simultaneously. This style—definition plus consequence in one breath—is the backbone of how formal mathematics still operates today: every concept nailed down before any reasoning can proceed.
Euclid taught in Alexandria around 300 BCE and compiled the Elements—thirteen books organizing all known Greek mathematics into an airtight axiomatic system. His defining trait was precision: no term used before it was defined, no claim made before it was proved. His famous reply to King Ptolemy I, that there is no royal road to geometry, reveals the same conviction: rigor is non-negotiable, even for rulers. This definition exemplifies that uncompromising standard.
Alexandria in 300 BCE was the intellectual center of the Hellenistic world, drawing scholars from across the Mediterranean to its great Library. Greek philosophers had elevated logical proof above all other forms of knowledge, yet no standardized mathematical vocabulary existed. Without agreed definitions, theorems built in Athens could not be verified in Carthage or Babylon. Euclid's methodical vocabulary—defining every term before using it—created the shared language that let mathematical knowledge accumulate reliably across generations and civilizations.
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