Euclid — "Let it be granted that all right angles are equal to one another."
Let it be granted that all right angles are equal to one another.
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More from Euclid
"A surface is that which has length and breadth only."
"The elements of geometry are derived from a small set of axioms and postulates."
"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."
"And when the lines containing the angle are straight, the angle is called rectilineal."
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Understanding this quote
What it means
This is one of Euclid's five foundational postulates — a starting assumption, not a proven fact. It declares that a right angle is universally constant: 90 degrees is the same regardless of where, when, or how you draw it. Rather than proving this obvious truth, Euclid acknowledged it must simply be accepted as given. This idea — that logical systems require unprovable starting assumptions — is now foundational to all mathematics and formal reasoning.
Relevance to Euclid
Euclid built all of geometry on just five postulates and five common notions — this is the fourth. His career was devoted to showing that rigorous knowledge could be constructed from minimal, self-evident truths through pure deduction. Teaching at Alexandria's Museum under Ptolemy I, he famously told the king there is no royal road to geometry. This postulate embodies his uncompromising intellectual honesty: name your assumptions plainly before claiming to prove anything.
The era
Around 300 BCE, Alexandria was the intellectual capital of the Hellenistic world, blending Greek philosophy with Egyptian scholarship. Aristotle had recently formalized deductive logic; Plato's Academy debated the nature of mathematical truth. Greek thinkers were pushing geometry from intuitive drawing toward rigorous proof. Euclid's postulates directly answered this moment — settling what must be assumed before proof begins, transforming centuries of informal mathematical practice into a permanent, universal logical foundation.
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