Euclid — "The extremities of a surface are lines."
The extremities of a surface are lines.
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More from Euclid
"If a straight line be drawn from the ends of a straight line, it will be a triangle."
"And when the lines containing the angle are straight, the angle is called rectilineal."
"Parallelograms which are on the same base and in the same parallels are equal to one another."
"That which is without parts has no magnitude."
"If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the …"
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Understanding this quote
What it means
A surface — any flat or curved 2D shape — ends at its edges, and those edges are lines. It's a precise geometric definition establishing that dimensions nest within each other: points bound lines, lines bound surfaces. If you draw a rectangle, its four sides are the lines marking where the surface stops. This hierarchical principle defines how shapes are constructed and bounded, forming a logical foundation for all geometric reasoning that follows.
Relevance to Euclid
This is Definition 6 from Euclid's Elements (~300 BCE), the axiomatic treatise that systematized all Greek geometry. Working in Alexandria, Euclid built every theorem from explicit definitions and postulates, allowing zero unstated assumptions. His career was dedicated precisely to this hierarchical decomposition — understanding complex shapes by reducing them to simpler elements. That surfaces terminate in lines reflects his core method: rigorous, incremental logical structure. His famous reply to Ptolemy — there is no royal road to geometry — shows how seriously he took this foundational discipline.
The era
Euclid worked around 300 BCE in Alexandria during the early Hellenistic period, when Ptolemy I was transforming the city into the Mediterranean's intellectual center and the Great Library was being established. Greek mathematics was shifting from Egyptian practical land-measurement toward abstract, proof-based systems. Plato's Academy had elevated geometry to philosophical status, but no unified text existed. Euclid filled that void, synthesizing predecessors like Eudoxus and Theaetetus. Precise definitions like this one were revolutionary — they replaced intuitive assumption with formal, unambiguous foundations.
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