Euclid — "And when the lines containing the angle are straight, the angle is called rectil…"
And when the lines containing the angle are straight, the angle is called rectilineal.
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Verification
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Understanding this quote
What it means
When two straight lines meet at a point, the space between them forms what geometry calls a rectilineal angle. This distinguishes angles made by straight lines from those made by curves. It's a foundational definition: before proving anything about angles, you must agree on what an angle is. The term 'rectilineal' simply means 'bounded by straight lines' — a precise label that prevents ambiguity in all subsequent geometric reasoning.
Relevance to Euclid
Euclid's entire legacy rests on the rigor of his definitions. The Elements opens with 23 definitions before stating a single postulate or theorem — a deliberate architecture. This quote is Definition 9 from Book I, exemplifying his method: distinguish every concept precisely before using it in proof. Euclid taught in Alexandria, likely at the Mouseion, and believed mathematics required an unshakeable logical foundation built entirely from agreed-upon terms.
The era
Around 300 BC, Greek mathematics was formalizing its break from Babylonian and Egyptian empiricism, where geometry served land surveying and construction. Plato's Academy had elevated geometry to philosophical ideal. Euclid wrote during Alexandria's early Ptolemaic era, when the city was becoming a center of scholarship. Defining terms precisely responded to earlier mathematical disputes — Greek thinkers had discovered that intuitive assumptions led to paradoxes, making rigorous definitions essential for trustworthy knowledge.
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