Euclidean geometry
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| + | "[[Gaston Bachelard]] showed that new theories integrate old theories in new [[paradigm]]s, changing the sense of concepts (for instance, the concept of [[mass]], used by [[Newton]] and [[Einstein]] in two different senses). Thus, [[non-Euclidean geometry]] did not contradict [[Euclidean geometry]], but integrated it into a larger [[framework]]."--Sholem Stein | ||
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| - | :[[Gaston Bachelard]] showed that new theories integrate old theories in new [[paradigm]]s, changing the sense of concepts (for instance, the concept of [[mass]], used by [[Newton]] and [[Einstein]] in two different senses). Thus, [[non-Euclidean geometry]] did not contradict [[Euclidean geometry]], but integrated it into a larger [[framework]]. | ||
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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]]. | '''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]]. | ||
| - | 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. | + | ==See also== |
| + | *[[Analytic geometry]] | ||
| + | *[[Type theory]] | ||
| + | *[[Interactive geometry software]] | ||
| + | *[[Non-Euclidean geometry]] | ||
| + | *[[Ordered geometry]] | ||
| + | *[[Incidence geometry]] | ||
| + | *[[Metric geometry]] | ||
| + | *[[Birkhoff's axioms]] | ||
| + | *[[Hilbert's axioms]] | ||
| + | *[[Parallel postulate]] | ||
| + | *[[Schopenhauer's criticism of the proofs of the Parallel Postulate]] | ||
| + | *[[Cartesian coordinate system]] | ||
| - | 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. | + | ===Classical theorems=== |
| + | *[[Ceva's theorem]] | ||
| + | *[[Heron's formula]] | ||
| + | *[[Nine-point circle]] | ||
| + | *[[Pythagorean theorem]] | ||
| + | *[[Menelaus' theorem]] | ||
| + | *[[Angle bisector theorem]] | ||
| + | *[[Butterfly theorem]] | ||
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"Gaston Bachelard showed that new theories integrate old theories in new paradigms, changing the sense of concepts (for instance, the concept of mass, used by Newton and Einstein in two different senses). Thus, non-Euclidean geometry did not contradict Euclidean geometry, but integrated it into a larger framework."--Sholem Stein |
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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.
[edit]
See also
- Analytic geometry
- Type theory
- Interactive geometry software
- Non-Euclidean geometry
- Ordered geometry
- Incidence geometry
- Metric geometry
- Birkhoff's axioms
- Hilbert's axioms
- Parallel postulate
- Schopenhauer's criticism of the proofs of the Parallel Postulate
- Cartesian coordinate system
[edit]
Classical theorems
- Ceva's theorem
- Heron's formula
- Nine-point circle
- Pythagorean theorem
- Menelaus' theorem
- Angle bisector theorem
- Butterfly theorem
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