Euclid — "A figure is that which is contained by any boundary or boundaries."
A figure is that which is contained by any boundary or boundaries.
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More from Euclid
"When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular …"
"A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle."
"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-angle…"
"To inscribe a regular hexagon in a given circle."
"Things which coincide with one another are equal to one another."
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Understanding this quote
What it means
A shape or form is defined entirely by whatever encloses it — its boundary is what makes it exist as a figure. Without a border, there is no shape, only open space. This is a foundational definition: before you can discuss triangles, circles, or polygons, you must agree on what a figure even is. The boundary is not decoration; it is the thing itself.
Relevance to Euclid
Euclid built all of geometry on precise definitions before proving anything — this is Definition 14 from Book I of his Elements, written around 300 BCE. His method of starting from irreducible definitions, then axioms, then proofs became the template for mathematical rigor for two millennia. That he defined 'figure' before tackling triangles or circles shows his commitment to leaving nothing assumed — the hallmark of his intellectual character.
The era
In Hellenistic Alexandria around 300 BCE, Greek thinkers were systematizing knowledge rather than merely applying it. Practical geometry had existed for centuries in Egypt and Mesopotamia for surveying and construction, but lacked logical foundations. Under Ptolemy I's patronage, the Library of Alexandria attracted scholars to organize inherited knowledge. Euclid's era demanded rigor: Plato had argued that only reasoned truth, not sensory observation, was reliable — making precise definitions philosophically necessary, not just mathematical housekeeping.
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