Euclid — "A line is breadthless length."
A line is breadthless length.
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More from Euclid
"To draw a straight line from any point to any point."
"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."
"The postulates are not self-evident, but they are necessary for the development of geometry."
"The square on the side subtending the right angle in right-angled triangles is equal to the squares on the sides containing the right angle."
"What advantage shall I get by learning these things?"
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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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