Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let all other quadrilaterals besides these be called trapezia.
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"Things which are equal to the same thing are also equal to one another."
"A number is a multitude composed of units."
"If a straight line be cut into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts together with the square on the line between the points of section is equal…"
"To draw a straight line at right angles to a given straight line from a given point on it."
"Let it be granted that a straight line may be drawn from any one point to any other point."
From 'Elements', Book I, Definition 23 (often summarized, this is the full definition)
Date: c. 300 BCE
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A square has four equal sides and four right angles. An oblong (rectangle) has right angles but unequal sides. A rhombus has equal sides but no right angles. A rhomboid has equal opposite sides and angles but neither equal sides nor right angles. Everything else is a trapezium. This is pure definitional precision — establishing shared vocabulary so geometry can reason without ambiguity.
Euclid built mathematics on explicit definitions before proving anything, a method central to his Elements. This passage is literally from Book I of Elements, where he lays foundational definitions. His genius was recognizing that geometry collapses into confusion without agreed-upon terms — so he defined everything first, reflecting his belief that rigorous foundations precede all reasoning.
Around 300 BCE in Alexandria, Greek scholars were systematizing knowledge across philosophy, medicine, and mathematics. No standardized geometric vocabulary existed before Euclid. Different Greek city-states used inconsistent terms. Euclid wrote under Ptolemy I, whose Library of Alexandria sought to gather and codify all human knowledge — making Euclid's definitional rigor perfectly aligned with that era's intellectual mission of universal systematization.
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