Euclid

:For other senses of this word, see Euclid (disambiguation).

The Elements

Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework. In addition to providing some missing proofs, Euclid's text also includes sections on number theory and three-dimensional geometry.

Related Topics:
Proof - Number theory - Three-dimensional geometry

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The geometrical system described in Elements was long known simply as "the" geometry. Today, however, it is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries which were discovered in the 19th century. These new geometries grew out of more than two millennia of investigation into Euclid's fifth postulate, one of the most-studied axioms in all of mathematics. Most of these investigations involved attempts to prove the relatively complex and presumably non-intuitive fifth postulate using the other four (a feat which, if successful, would have shown the postulate to be in fact a theorem).

Related Topics:
Euclidean geometry - ''non-Euclidean'' geometries - 19th century - Millennia - Fifth postulate - Axiom - Theorem

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While the Elements was used well into the 20th century as a geometry textbook and has been considered a fine example of the formally precise axiomatic method, Euclid's treatment does not hold up to modern standards of rigor; some logically necessary axioms are missing, and the definitions of primitive terms appeal to spatial intuition. The first correct axiomatic treatment of geometry by modern standards was provided by David Hilbert in 1899, in his Grundlagen der Geometrie.

Related Topics:
20th century - David Hilbert - 1899

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