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Euclidean Geometry

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The study of the magnitudes and characteristics of the figures found in space or in a plane is called geometry. Euclidean, meanwhile, is that linked to Euclid, a mathematician who lived in Ancient Greece. And not only that but also that this illustrious figure became a teacher of important disciples such as Apollonius of Perga or Archimedes, among many others.

In the third century BC, Euclid proposed five postulates that allow studying the properties of regular forms (lines, triangles, circles, etc.). Thus gave birth to Euclidean geometry.

At present, Euclidean geometry is considered to be one that focuses on the analysis of the properties of Euclidean spaces: the geometric spaces that meet the axioms of the Greek thinker. It should be noted that Euclides compiled his postulates in his work "Elements".

In this treatise, Euclid points out that a straight line can be created from the union of any two points; that a segment of a line can extend indefinitely in a straight line; that, given a line segment, a circle with any distance and center can be drawn; that all right angles are identical to each other; and that, if a straight line cuts to two others and the sum of the interior angles of the same side is less than two right angles, the other two straight lines will be cut by the side where the angles are less than the right angles.

When working with Euclidean spaces, Euclidean geometry takes care of complete vector spaces that have an internal product and, therefore, are metric and standard vector spaces. The spaces of the non-Euclidean geometries, on the other hand, are curved spaces or with different characteristics to those mentioned in Euclid's propositions.

From this work entitled 'Elements' we must establish other interesting data, among which we can highlight that it's composed of thirteen books, that it was the masterpiece of its author and that focuses on treating the geometry of both two dimensions and three dimensions.

Likewise, it must be taken into account that it's considered one of the most edited works in history, since it has more than a thousand editions. However, one of the most interesting editions, without a doubt, is the one carried out by Archimedes of Syracuse.

In addition to all these data there are others that must also be considered:

- All proposals or postulates are presented axiomatically.
- It didn't begin to spread and to stand out in Europe until the Low Average Age.
- For the scientific community it became an essential work and it was for many centuries. Specifically, until the appearance of the theory of relativity of Albert Einstein.
- The structure of this work is as follows: books 1 to 4 focus on flat geometry, books 5 to 10 revolve around what are the proportions and reasons while the last three books address what is the geometry of the three dimensions, the geometries in the bodies that are solid.


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