Euclidean geometry
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| + | '''Euclidean geometry''' is a mathematical system attributed to the [[Greeks|Greek]] [[mathematician]] [[Euclid]] of [[Alexandria]]. Euclid's text ''[[Euclid's Elements|Elements]]'' is the earliest known systematic discussion of [[geometry]]. It has been one of the most influential books in history, as much for its method as for its mathematical content. The method consists of assuming a small set of intuitively appealing [[axiom]]s, and then proving many other [[proposition]]s ([[theorem]]s) from those axioms. Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fit together into a comprehensive deductive and [[logical system]]. | ||
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| + | The ''Elements'' begin with [[plane geometry]], still taught in [[secondary school]] as the first [[axiomatic system]] and the first examples of [[Mathematical proof|formal proof]]. The ''Elements'' goes on to the [[solid geometry]] of three [[dimension]]s, and Euclidean geometry was subsequently extended to any finite number of [[dimension]]s. Much of the ''Elements'' states results of what is now called [[number theory]], proved using geometrical methods. | ||
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| + | For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute sense. Today, however, many other [[self-consistent]] [[non-Euclidean geometry|non-Euclidean geometries]] are known, the first ones having been discovered in the early 19th century. It also is no longer taken for granted that Euclidean geometry describes physical space. An implication of [[Einstein]]'s theory of [[general relativity]] is that Euclidean geometry is a good approximation to the properties of physical space only if the [[gravity|gravitational field]] is not too strong. | ||
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Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content. The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions (theorems) from those axioms. Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fit together into a comprehensive deductive and logical system.
The Elements begin with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. The Elements goes on to the solid geometry of three dimensions, and Euclidean geometry was subsequently extended to any finite number of dimensions. Much of the Elements states results of what is now called number theory, proved using geometrical methods.
For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. It also is no longer taken for granted that Euclidean geometry describes physical space. An implication of Einstein's theory of general relativity is that Euclidean geometry is a good approximation to the properties of physical space only if the gravitational field is not too strong.
