Euclid — "Rectilineal figures are those which are contained by straight lines..."
Rectilineal figures are those which are contained by straight lines...
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More from Euclid
"In any right-angled triangle, the square on the side subtending the right angle is equal to the squares on the sides containing the right angle."
"A boundary is that which is an extremity of anything."
"That which is without parts has no magnitude."
"Let it be granted that a circle may be described with any center and any radius."
"To produce a finite straight line continuously in a straight line."
Verification
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Understanding this quote
What it means
Shapes made entirely of straight edges — triangles, squares, polygons — belong to one category; curved shapes like circles belong to another. Before proving anything, every term must be exact. This is a geometric definition: rectilineal figures are simply what we now call polygons. The precision is the point. Nothing is left assumed or vague; the classification must be airtight before any logical reasoning can proceed on top of it.
Relevance to Euclid
Euclid built Elements around 23 definitions before stating a single axiom or proving any theorem — this quote is Definition 19. His defining trait was methodical rigor: nothing assumed, everything stated explicitly. Working in Alexandria, he synthesized centuries of Greek geometric knowledge into one unified deductive system. The definition reflects his core belief that mathematics must begin with airtight foundations; only from exact language can reliable, universal proofs follow.
The era
Euclid worked in Alexandria around 300 BCE during the early Ptolemaic dynasty, when the Great Library was becoming the ancient world's intellectual center. Greek thinkers had long used geometric reasoning informally, but no one had organized it axiomatically. Plato's Academy had elevated geometry as the highest intellectual discipline. Euclid's era demanded rigor over intuition — this definition embodies that cultural drive to ground all knowledge in first principles before building further.
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