Understanding Euclidean Geometry

The ancient Greek mathematician Euclid is credited with developing a mathematical system that is now known as Euclidean geometry. Euclid defined this method in his geometry textbook, which was titled the Elements. Euclid’s method involves starting with a relatively limited number of axioms that are intuitively appealing and then deducing a large number of further propositions (theorems) from these. Euclid was the first person to arrange these statements into a logical system in which each result is proved from axioms and previously demonstrated theorems. Although many of Euclid’s conclusions had been stated before, Euclid was the first person to organise these propositions into a logical system. 

The book begins with plane geometry, which is still taught in secondary schools (high schools) as the first axiomatic system and the earliest instances of mathematical proofs. The Elements was written by the Greek mathematician Euclid. The discussion then moves on to the three-dimensional solid geometry. The conclusions of what is now known as algebra and number theory are discussed throughout most of the Elements using terminology that is exclusive to geometry. 

Because no other type of geometry had been conceived of during that time period, the term “Euclidean” was superfluous for more than two thousand years. With the probable exception of the parallel postulate, Euclid’s axioms appeared to be so intuitively evident that every theorem established from them was thought to be true in an absolute, often metaphysical sense. The earliest examples of self-consistent non-Euclidean geometries were found in the early 19th century; however, there are now a great number of different non-Euclidean geometries that are known. In Albert Einstein’s theory of general relativity, one of the implications is that physical space itself is not Euclidean, and that Euclidean space is only a good approximation for it over short distances. In other words, Euclidean space is only a good approximation when it is close to physical space (relative to the strength of the gravitational field). 

The branch of geometry known as Euclidean geometry is an example of synthetic geometry since it follows a logical progression from axioms, which describe fundamental qualities of geometric objects like points and lines, through propositions regarding those geometric objects. On the other hand, analytic geometry, which was developed approximately 2,000 years later by René Descartes and involves the use of coordinates to describe geometric characteristics as algebraic formulae, stands in contrast to this.

The field of Euclidean geometry is known as an axiomatic system because all of its theorems, also known as “true statements,” are derived from a limited number of fundamental principles. These axioms were thought to be self-evidently correct in the real world up to the development of non-Euclidean geometry, which meant that all the theorems were assumed to have the same level of validity. Euclid’s argument from assumptions to conclusions, on the other hand, holds true even when their physical actuality is taken into consideration. 

Near the beginning of the first book of the Elements, Euclid lays down five postulates (axioms) for plane geometry. These postulates are presented in terms of constructions (according to Thomas Heath’s translation):

 The following hypothesis should be considered:

1. To connect any two points using a line that is perfectly straight.
2. To create (stretch) a finite straight line by proceeding in an unbroken straight line.
3. To characterise a circle with a certain centre and distance from that centre (radius).
4. That each and every right angle is equivalent to the others.
5. [The parallel postulate] states that if a straight line falling on two other straight lines makes the interior angles on the same side less than two right angles, then the two straight lines will meet on the side on which the angles are less than two right angles if they are produced indefinitely.

Euclid makes an implicit assumption that the created things are one of a kind in his reasoning, despite the fact that he only expressly claims that they exist as constructed objects.

In addition to this, the Elements consist of the five “common concepts” listed below:

1. Those things are also equal to one another if they are equal to the object that they are equal to (the transitive property of a Euclidean relation).
2. If you add equals to equals, you will find that the wholes are also equal (Addition property of equality).
3. If one set of equals is subtracted from another set of equals, then the differences are also equal (subtraction property of equality).
4. Things that are equivalent to one another are those that correspond with one another (reflexive property).
5. The sum is higher than the components that make it up.

Plane

A flat, two-dimensional surface that extends endlessly in all directions is referred to as a plane in mathematics.

 A point has zero dimensions, a line has one dimension, and three-dimensional space has three dimensions. A plane, on the other hand, only has two dimensions. Planes can either originate as subspaces of some higher-dimensional space, such as when one of the walls of a room is indefinitely extended, or they can have their own independent existence in their own right, such as when two-dimensional Euclidean geometry is being used.

When working just in two-dimensional Euclidean space, the definite article is utilised, and the plane is understood to refer to the entirety of the space being worked in. In the fields of mathematics, geometry, trigonometry, graph theory, and graphing, many of the most fundamental activities are done in a two-dimensional environment, most frequently in the plane.

Properties

The following statements are true in three-dimensional Euclidean space but not in higher dimensions; however, they do have analogues in higher dimensions:

• Two distinct planes are either parallel to each other or they intersect in a line
• A line is either parallel to a plane, intersects it at a single point, or is contained in the plane
• Two distinct lines perpendicular to the same plane must be parallel to each other

Conclusion:

The study of solid and flat objects using the axioms and theorems developed by the Greek mathematician Euclid is known as Euclidean geometry (c. 300 BCE). Euclidean geometry, in broad strokes, is the solid and plane geometry often taught in secondary schools. Geometry referred to Euclidean geometry up to the second half of the 19th century, when non-Euclidean geometries caught the interest of mathematicians. It is the most prevalent example of how people generally think mathematically. It requires real insight into the subject, creative ideas for applying theorems in unique situations, the capacity to generalise from known facts, and a stress on the necessity of evidence rather than the memorising of straightforward algorithms to solve equations by rote. The only tools used for geometrical structures in Euclid’s famous book, the Elements, were the ruler and the compass; this limitation is still present in basic Euclidean geometry.