Euclidean Geometry

The study of geometrical shapes (plane and solid) and figures based on various axioms and theorems is known as Euclidean geometry. It is primarily intended for use on flat or plane surfaces. Geometry is derived from the Greek terms ‘geo’ (meaning earth) and ’metrein’ (meaning ‘to measure’).

The shapes of geometrical objects and planes are better explained using Euclidean geometry. The Greek mathematician Euclid used this aspect of geometry, which he also explained in his book Elements. As a result, this geometry is sometimes referred to as Euclid geometry.

The axioms, also known as postulates, are unproven assumptions that are clear universal truths. In his book Elements, Euclid established the principles of geometry, such as geometric shapes and figures, and defined five main axioms or postulates. The definition of euclidean geometry, its constituents, axioms, and five important postulates will be discussed here.

History of Euclidean Geometry:-

The Indus Valley Civilization’s very well-planned settlements are depicted in the excavations at Harappa and Mohenjo-Daro (about 3300-1300 BC). The Egyptians’ perfect construction of pyramids is yet another illustration of the people’s broad usage of geometrical skills during the time. The Sulba Sutras, as well as texts on geometry in India, show that the Indian Vedic Period had a Geometry heritage.

When Euclid, a teacher of mathematics in Alexandria, Egypt, collected most of these evolutions in geometry and organised them into his renowned book, which he named ‘Elements,’ the progress of geometry was slow.

Meaning:-

All the theorems of Euclidean Geometry are derived from a small number of simple axioms, making it an axiomatic system. Euclidean Geometry is also known as “planar geometry” since it deals with objects like points, lines, angles, squares, triangles, and other shapes. It is concerned with all objects’ qualities and interactions.

Example of Euclidean Geometry:-

Angles and circles are two often used examples of Euclidean geometry. The inclination of two straight lines is referred to as an angle. A circle is a flat figure with all of its points at a fixed distance from the centre (called the radius).

Euclidean and Non-Euclidean Geometry:-

In terms of parallel lines, there is a distinction between Euclidean and non-Euclidean geometry. In Euclidean geometry, there is precisely one line that travels through the provided points in the same plane and never intersects for every given point and line.

Non-Euclidean geometry is not the same as Euclidean geometry. Because the lines in spherical geometry are not straight, it is a non-Euclidean geometry.

Properties of Euclidean Geometry:-

It entails the study of both plane and solid geometry.

It established a point, a line, and a plane.

A solid possesses shape, size, and position, as well as the ability to move from one location to another.

The sum of all three angles of any type of triangle gives 180° .

Two parallel lines never cross. The straight line is always the shortest distance between any two points.p

Elements in Euclidean Geometry:-

Euclid’s Elements is a mathematical and geometrical work produced by ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt, consisting of 13 volumes. In addition, the ‘Elements’ were grouped into thirteen books, which popularized geometry around the world. These Elements are made up of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions as a whole.

Plane geometry is discussed from the first through the fourth and sixth books. Euclid’s Postulates are five postulates for plane geometry, and the geometry is known as Euclidean geometry. We have a communal source for studying geometry thanks to his work; it provides the foundation for geometry as we know it today.

Euclidean Axioms:-

Euclid’s seven axioms for geometry are listed here;

Things which are equal to the same thing are equal to one another. 

If equals are added to equals, the wholes are equal. 

If equals are subtracted from equals, the remainders are equal. 

Things which coincide with one another are equal to one another. 

The whole is greater than the part. 

Things which are double of the same things are equal to one another. 

Things which are halves of the same things are equal to one another

Euclidean Postulates:-

The below following are the postulates as given by Euclid;

A straight line may be drawn from any point to another point.

A terminated line can be produced indefinitely.

A circle can be drawn with any centre and radius.

All right angles are equal to each other.

If a straight line falls on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

Conclusion:-

Euclidean geometry is the study of planar and solid forms based on the axioms and theorems of Euclid, a Greek mathematician (c. 300 BCE). Euclidean geometry, in its most basic form, is the plane and solid geometry frequently taught in secondary schools. Indeed, until the second half of the nineteenth century, when mathematicians became interested in non-Euclidean geometries, geometry meant Euclidean geometry. It is the most common example of generic mathematical reasoning.