Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.
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"If four magnitudes be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means."
"To draw a straight line at right angles to a given straight line from a given point on it."
"A point is that which has no part."
"Rectilineal figures are those which are contained by straight lines..."
"And that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on …"
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Shapes enclosed entirely by straight lines are classified by how many lines bound them: three lines make a triangle, four a quadrilateral, and five or more a multilateral polygon. This is pure definitional geometry — no measurement, no calculation, just precise categorization by structure. Every polygon you recognize today — triangles, squares, pentagons — traces directly back to this foundational sorting by side count.
Euclid built his entire Elements on exactly this kind of meticulous definition — establish every term precisely before proving anything. Teaching in Alexandria around 300 BCE, his genius wasn't discovering new shapes but constructing a logical architecture where nothing was assumed. This definition captures his character: rigorous, systematic, stripping concepts to their irreducible essence so that every theorem built on top could rest on unshakeable ground.
In 300 BCE Alexandria, the Ptolemaic empire was positioning itself as heir to Greek intellectual culture, with the great Library centralizing knowledge. Plato and Aristotle had already established that philosophy and science demanded unambiguous foundational terms. Euclid's definitional rigor also answered real disputes — without fixing exactly what 'triangle' meant, competing proofs could be challenged. Precise language was not pedantry; it was the prerequisite for trustworthy reasoning.
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