Euclid — "The extremities of a line are points."
The extremities of a line are points.
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More from Euclid
"That all right angles are equal to one another."
"And the point is called the center of the circle."
"A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line."
"Trilateral figures are those contained by three straight lines, quadrilateral those contained by four, and multilateral those contained by more than four straight lines."
"Let it be granted that a finite straight line may be produced to any length in a straight line."
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Understanding this quote
What it means
A line extends continuously, but wherever it ends or terminates, that boundary is a point — geometry's most fundamental unit. This is a precise definitional statement: points have no dimension, lines have one. The quote establishes that a line's limits are not vague regions but exact, dimensionless locations. It's foundational logic — you must define your terms rigorously before building any system of knowledge on top of them.
Relevance to Euclid
This is literally Definition 3 from Book I of Euclid's Elements, written around 300 BCE in Alexandria. Euclid built all of geometry from minimal assumptions — definitions, postulates, axioms — then derived everything through logical proof. That methodical precision is his identity. Defining the endpoint of a line before proving anything about lines reflects his conviction that rigorous foundations, not intuition, must underpin mathematics. His Elements remained the standard geometry textbook for over 2,000 years.
The era
Around 300 BCE, Ptolemy I had established Alexandria as a Mediterranean intellectual hub anchored by its great Library. Greek thinkers, shaped by Plato and Aristotle, were formalizing abstract reasoning across philosophy and science. Mathematics was shifting from practical land-surveying toward pure logical inquiry. Defining geometric terms rigorously mattered because competing philosophical schools disputed what constituted valid knowledge — Euclid's axiomatic method answered that dispute with a system that felt unassailable for millennia.
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