Euclidean Geometry

Pronunciation: /yuˈklɪ diən dʒiˈɒm ɪ tri/ ?

Euclidean geometry is the geometry described by the mathematician Euclid (ca 300 BC) in his landmark work Elements.[1] As head librarian at the famed Library of Alexandria, Euclid had access to the best minds of the ancient world.

Euclid's main contribution to mathematics was an axiomatic approach. The axiomatic approach involves basic statements called axioms that are taken to be true without proof. All other conclusions are proved from these axioms.

In his book Elements, Euclid described five axioms:

  1. Any two points can be joined by a straight line.
  2. Any straight line segment can be extended indefinitely in a straight line.
  3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
  4. All right angles are congruent.
  5. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
From these five axioms, all other theorems in Euclidean geometry are proved. In general, Euclidean geometry is distinguished from non-Euclidean geometry by the fifth postulate, also called the parallel postulate. In attempting to prove that the fifth postulate was not necessary to Euclidean geometry, mathematicians discovered that, if the fifth postulate was not included, other geometries different from Euclidean geometry were possible.


  1. Euclidean geometry. Encyclopedia Britannica. (Accessed: 2010-01-25). geometry.
  2. Casey, John, LL.D., F.R.S.. The First Six Books of the Elements of Euclid. Casey, John, LL.D. F.R.S.. Hodges, Figgis & Co., 1890. (Accessed: 2010-01-26).

More Information

  • Euclid. Elements. 2009-04-19.

Printed Resources

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Euclidean Geometry. 2010-01-26. All Math Words Encyclopedia. Life is a Story Problem LLC.


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2010-01-26: Added "References" (McAdams, David.)
2007-07-12: Initial version (McAdams, David.)

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