Euclidean Geometry

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, couched in geometrical language.

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 (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Einstein's theory of general relativity is that Euclidean space is a good approximation to the properties of physical space only where the gravitational field is weak.

Read more about Euclidean Geometry:  The Elements, Methods of Proof, System of Measurement and Arithmetic, Some Important or Well Known Results, Applications, As A Description of The Structure of Space, Logical Basis

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List Of First-order Theories - Geometry
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... The process of abstract axiomatization as exemplified by Hilbert's axioms reduces geometry to theorem proving or predicate logic ... Andrei Nicholaevich Kolmogorov proposed a problem solving basis for geometry ...
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... Although there was a prevailing Kantian consensus at the time, once non-Euclidean geometries had been formalised, some began to wonder whether or not physical space is curved ... in which he attempted to demonstrate the futility of any attempt to discover which geometry applies to space by experiment ... for the debate over whether real space is Euclidean or not ...
Coordinative Definition - Formalism
... For the formalists, mathematics, and particularly geometry, is divided into two parts the pure and the applied ... basis of deductive rules eternally specified in advance, pure geometry provides a set of theorems derived in a purely logical manner from the axioms ... Reichenbach's arguments about the nature of geometry on the one hand, we cannot know if there are universal forces until we know the true geometry of spacetime, but on the other, we ...
Models Of Non-Euclidean Geometry
... Models of non-Euclidean geometry are mathematical models of geometries which are non-Euclidean in the sense that it is not the case that exactly one line can be drawn parallel to a given line l through a point ... See the entries on hyperbolic geometry and elliptic geometry for more information.) Euclidean geometry is modelled by our notion of a "flat plane." The simplest model for elliptic geometry is a sphere ... curvature to model hyperbolic geometry ...

Famous quotes containing the word geometry:

    I am present at the sowing of the seed of the world. With a geometry of sunbeams, the soul lays the foundations of nature.
    Ralph Waldo Emerson (1803–1882)