Euclidean geometry is a mathematical system attributed to Alexandrian

7 min readMay 29, 2021

Euclidean geometry is a mathematical system attributed to 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,[1] Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system.[2] The Elements begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. 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, explained in geometrical language.[1]

For more than 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 Albert Einstein’s theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances (relative to the strength of the gravitational field).[3]

Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects. This is in contrast to analytic geometry, which uses coordinates to translate geometric propositions into algebraic formulas.

Contents
1 The Elements
1.1 Axioms
1.2 Parallel postulate
2 Methods of proof
3 System of measurement and arithmetic
4 Notation and terminology
4.1 Naming of points and figures
4.2 Complementary and supplementary angles
4.3 Modern versions of Euclid’s notation
5 Some important or well known results
5.1 Pons Asinorum
5.2 Congruence of triangles
5.3 Triangle angle sum
5.4 Pythagorean theorem
5.5 Thales’ theorem
5.6 Scaling of area and volume
6 Applications
7 As a description of the structure of space
8 Later work
8.1 Archimedes and Apollonius
8.2 17th century: Descartes
8.3 18th century
8.4 19th century and non-Euclidean geometry
8.5 20th century and relativity
9 Treatment of infinity
9.1 Infinite objects
9.2 Infinite processes
10 Logical basis
10.1 Classical logic
10.2 Modern standards of rigor
10.3 Axiomatic formulations
11 See also
11.1 Classical theorems
12 Notes
13 References
14 External links
The Elements
Main article: Euclid’s Elements
The Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was rapidly recognized, with the result that there was little interest in preserving the earlier ones, and they are now nearly all lost.

There are 13 books in the Elements:

Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, for example, “In any triangle two angles taken together in any manner are less than two right angles.” (Book I proposition 17) and the Pythagorean theorem “In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.” (Book I, proposition 47)

Books V and VII–X deal with number theory, with numbers treated geometrically as lengths of line segments or areas of regions. Notions such as prime numbers and rational and irrational numbers are introduced. It is proved that there are infinitely many prime numbers.

Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. The platonic solids are constructed.

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Euclidean geometry is a mathematical system attributed to Alexandrian

Written by Mafor