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Home Mathematics Plane Geometry Euclid’s five postulates

Euclid’s five postulates

Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid.


Euclid's Elements is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions.

Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions.

"Let the following be postulated":

  1. "To draw a straight line from any point to any point."
  2. "To produce [extend] a finite straight line continuously in a straight line."
  3. "To describe a circle with any centre and distance [radius]."
  4. "That all right angles are equal to one another."
  5. The parallel postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."
Although Euclid's statement of the postulates only explicitly asserts the existence of the constructions, they are also taken to be unique.

The five postulates may be translated into the following :

  1. Two points determine a straight line.
  2. A line segment extended infinitely in both directions produces a straight line.
  3. A circle is determined by a center and distance.
  4. All right angles are equal to one another.
  5. If a straight line falling an two straight lines forms interior angles on the same side less than 180 degrees, the two straight lines, if produced indefinitely, will meet on that side.

The Elements also include the following five "common notions":

  1. Things that are equal to the same thing are also equal to one another.
  2. If equals are added to equals, then the wholes are equal.
  3. If equals are subtracted from equals, then the remainders are equal.
  4. Things that coincide with one another equal one another.
  5. The whole is greater than the part.

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