*Macmillan Science Library: Mathematics*. Copyright © 2001-2006 by Macmillan Reference USA, an imprint of the Gale Group. All rights reserved.

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## Negative Discoveries

While solving problems and constructing proofs, mathematicians use many different approaches. A common technique for proving a statement is by contradiction. In this approach, it is supposed that the converse of the statement, or its opposite, is true. If this supposition leads to an absurd result, or contradiction, then it can be said the original statement is true. Hence, exploring incorrect answers and assumptions can often lead to new correct results.

## Euclidean and Non-Euclidean Geometry

In 300 B.C.E., Euclid of Alexandria put forward a logical construction of a geometry, which has come to be known as Euclidean geometry. Until the middle of the nineteenth century mathematicians believed that Euclid's geometry was the only type of geometry possible. Euclidean geometry is based on a number of fundamental statements called postulates, or axioms.

In his book *Elements,* Euclid based his geometry on five axioms. The fifth axiom...

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