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What are Euclidâs five postulates?
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Euclid
Euclid, coloured woodcut, 1584. His Elements, a treatise on geometry and mathematics, is one of the most influential books ever written.
In his foundational work, the Elements, the Greek mathematician Euclid established five intuitive principles, known as axioms or postulates, to serve as the basis for his system of geometry. These are stated as follows:
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- Given two points, there is a straight line that joins them.
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- A straight line segment can be prolonged indefinitely.
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- A circle can be constructed when a point for its center and a distance for its radius are given.
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- All right angles are equal.
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- If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which the angles are less than the two right angles.
The fifth principle is famously known as the parallel postulate. Because it was considered less self-evident than the others, it eventually led mathematicians to explore non-Euclidean geometries.1