 # Introduction To Euclidean Geometry

Euclidean geometry is the study of geometry (especially for the shapes of geometrical figures) which is attributed to the Alexandrian mathematician Euclid who has explained in his book on geometry known as Elements. This geometry is also called Euclid geometry. The axioms or postulates are the assumptions which are obvious universal truths, they are not proved. Euclid has introduced the geometry fundamentals like geometric shapes and figures in his book elements and has stated 5 main axioms or postulates.

## Euclidean Geometry in Mathematics

 Fundamentals Plane geometry Solid geometry

## Introduction to Euclid Geometry

The excavations at Harappa and Mohenjo-Daro depict the extremely well-planned towns of Indus Valley Civilization (about 3300-1300 BC). The flawless construction of Pyramids by the Egyptians is yet another example of extensive use of geometrical techniques used by the people back then. In India, the Sulba Sutras, textbooks on Geometry depict that the Indian Vedic Period had a tradition of Geometry.

The development of geometry was taking place gradually, when Euclid, a teacher of mathematics, at Alexandria in Egypt, collected most of these evolutions in geometry and compiled it into his famous treatise, which he named ‘Elements’.

## What is Euclidean Geometry?

Euclidean Geometry is considered as an axiomatic system, where all the theorems are derived from the small number of simple axioms. Since the term “Geometry” deals with things like points, line, angles, square, triangle, and other shapes, the Euclidean Geometry is also known as the “plane geometry”. It deals with the properties and relationship between all the things. Non-Euclidean is different from Euclidean geometry. There is a difference between these two in the nature of parallel lines. In Euclid geometry, for the given point and line, there is exactly a single line that passes through the given points in the same plane and it never intersects.

## Euclid’s Elements

Euclid’s Elements is a mathematical and geometrical work consisting of 13 books written by ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt. Further, the ‘Elements’ was divided into thirteen books which popularized geometry all over the world. As a whole, these Elements is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions.

Book 1 to 4th and 6th discuss plane geometry. He gave five postulates for plane geometry known as Euclid’s Postulates and the geometry is known as Euclidean geometry. It was through his works, we have a collective source for learning geometry; it lays the foundation for geometry as we know now.

## Euclidean Geometry Axioms

1. Things are equal to one another if those things are equal to the same thing.
2. The wholes are equal if equals are added to equals.
3. The remainders are equal if equals are subtracted from equals.
4. Things are equal to one another if they coincide with one another.
5. The whole will be greater than the part.

## Euclid’s Five Postulates

Before discussing Euclid’s Postulates let us discuss a few terms as listed by Euclid in his book 1 of the ‘Elements’. The postulated statements of these are:

• Assume the three steps from solids to points as solids-surface-lines-points. In each step, one dimension is lost.
• A solid has 3 dimensions, the surface has 2, the line has 1 and point is dimensionless.
• A point is anything that has no part, a breadthless length is a line and the ends of a line point.
• A surface is something which has length and breadth only.

It can be seen that the definition of a few terms needs extra specification. Now let us discuss these Postulates in detail.

### Euclid’s Postulate 1

“A straight line can be drawn from anyone point to another point.”

This postulate states that at least one straight line passes through two distinct points but he did not mention that there cannot be more than one such line. Although throughout his work he has assumed there exists only a unique line passing through two points. ### Euclid’s Postulate 2

“A terminated line can be further produced indefinitely.”

In simple words what we call a line segment was defined as a terminated line by Euclid. Therefore this postulate means that we can extend a terminated line or a line segment in either direction to form a line. In the figure given below, the line segment AB can be extended as shown to form a line. ### Euclid’s Postulate 3

“A circle can be drawn with any center and any radius.”

Any circle can be drawn from the end or start point of a circle and the diameter of the circle will be the length of the line segment.

### Euclid’s Postulate 4

“All right angles are equal to one another.”

All the right angles (i.e. angles whose measure is 90°) are always congruent to each other i.e. they are equal irrespective of the length of the sides or their orientations.

### Euclid’s Postulate 5

“If a straight line falling on two other 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 the side on which the sum of angles is less than two right angles.”