Euclid — "A plane angle is the inclination of the lines to one another, when two lines mee…"
A plane angle is the inclination of the lines to one another, when two lines meet one another, but are not in the same straight line.
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More from Euclid
"A point is that which has no part."
"And the point is called the center of the circle."
"What has been affirmed without proof can also be denied without proof."
"A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another."
"Let it be granted that a circle may be described with any center and any radius."
Verification
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Understanding this quote
What it means
An angle is formed when two lines meet at a point but don't continue in the same direction. The 'inclination' is simply how much one line tilts away from the other at their meeting point. This is geometry's foundational definition of angle — precise, stripped of assumption, built only from observable properties of intersecting lines without invoking measurement or degrees.
Relevance to Euclid
Euclid wrote this in Elements around 300 BCE, his systematic compilation of geometry. His method — defining every term before using it — was revolutionary rigor. This definition reflects his obsession with building mathematics from undeniable first principles. He trusted no prior assumption, instead constructing the entire edifice of geometry from definitions like this one upward.
The era
In Alexandria around 300 BCE, Greek thinkers were systematizing knowledge under Ptolemaic patronage. Mathematics existed but lacked unified logical foundations. Euclid synthesized centuries of scattered geometric knowledge into a deductive system. Defining 'angle' precisely mattered because competing informal definitions caused contradictions. His Elements would dominate mathematical education for over 2,000 years, shaping every subsequent civilization's approach to reasoning.
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