Euclid — "A line is breadthless length."
A line is breadthless length.
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More from Euclid
"A boundary is that which is an extremity of anything."
"That all right angles are equal to one another."
"There are infinitely many prime numbers."
"The extremities of a line are points."
"If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the …"
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Understanding this quote
What it means
A line has only one dimension: length. It has no width, no thickness, no physical presence — it is purely an abstraction. This strips away all sensory intuition about what a line looks like in the real world and defines it with logical precision, establishing only what is mathematically necessary. It forces the mind to work with pure concept rather than observable reality.
Relevance to Euclid
Euclid wrote this as Definition 2 in his foundational work Elements, compiled around 300 BCE. His entire project was building geometry from irreducible definitions and axioms upward through rigorous proof. This definition reflects his systematic, uncompromising commitment to logical foundations — he refused to assume anything, even something as obvious as what a line is, without first stating it explicitly.
The era
In ancient Alexandria around 300 BCE, Greek thinkers were separating mathematical truth from physical measurement. Egyptian and Babylonian traditions used geometry practically — for land surveying and construction. Euclid represented a distinctly Greek intellectual shift: mathematics as a system of pure reasoning, independent of physical tools or measurement. Defining a line abstractly rather than practically marked this philosophical transformation.
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