Introduction Euclidean geometry and it's Elements

Geometry is developed from the time of human civilization. The tradition of the development is basically rooted in three ancient civilizations. These civilizations are (a) Babylonian, (b) Egypt and (c) Greek. There are number of reasons to evident these civilization as root of geometry development. Among them (a) trade route of Babylonian, (b) Nile river of Egypt and (c) “Geo + Metron” version of geometry in Greek are vital.

Along with three ancient civilization, there were number of mathematicians who contributed for the development of Geometry. Among them, following contributions are significantly counted.

1. Demonstrative approach of Thales (600 BC)
2. Pythagoras theorem of Pythagoras (500 BC)
3. Elements of geometry of Hippocrates-(400 BC)
4. Academy of Plato (400 BC):
5. Axiomatization of Eudoxus (300 BC)

Comprising the early development of Geometry dated back to 300BC, a Greek mathematician Euclid had systematically organized (compiled) Elements in 300BC. This Elements is known as Geometry. This is also known as Euclidean Geometry. In this way, formal birth of geometry get by Euclid in 300BC.

By that time, Euclid defined the geometry as “Geo+Metron” or “measuring earthly things”. In his speculation, the geometry was based on flat surface. Therefore, it is also called plane geometry. However, these days, the meaning of Geometry has been widen. These days, Geometry can be defined in different perspectives. Among them, the four common perspectives are mentioned below.

1. One of the mother structure of Mathematics
2. Study of two properties
3. Study of five invariants
4. An axiomatic System

Elements

Around 300 BC, a Greek mathematician Euclid compiled volume of 13 books. These books were a systemic organization of geometrical knowledge developed by that time. The volume of these 13 books is called Elements. The Elements consists of following major components.

1. Five common notions
2. Five postulates
3. 118 definitions
4. 465 propositions

The five common notions are

1. Things equal to same thing are equal.
2. If equals are added to equals, the result is equal.
3. If equals are subtracted from equals, the result is equal.
4. Things coincide to each other are equal.
5. The whole is greater than the part.

The five postulates are

1. A straight line can be drawn between two points.
2. A straight line can be extend continuously in a straight line.
3. A circle can be describe with a point and a distance.
4. All right angles are equal.
5. If a straight line is falling on two straight lines, make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.

In addition to this, the details of Elements are given below.

SN	C	P	D	T	C=common notions, P=Postulates, D=Definitions, T=Propositions (Theorems)
1	5	5	23	48	Basic Geometry
2			2	14	Geometric algebra
3			11	37	Circles
4			7	16	Construction
5			18	25	Theory of abstract proportions
6			4	33	Similarity
7			22	39	Fundamental of number theory
8				27	Continued proportions in number theory
9				36	Advances in number theory
10			3	115	Incommensurable number
11			28	39	Solid geometry
12				18	Mensuration
13				18	Regular solids

Incommensurable number:

Two numbers are called Incommensurable number if their ratio is an irrational number.

a

a

Regular solid

A polygon whose all interior angles are equal, all sides are equal, is called regular polygon. For example, square is a regular polygon.

There are infinite number of regular polygon that exist.

A solid whose all sides (edges) are equal, all faces are equal, is called regular solid. For example, cube is a regular solid.

There are only five regular solids that exist in the universe. This solid were introduced by Plato in 400BC. Therefore, it is also called Platonic solids. These solids are given as below.

SN	Name	V	E	F	V-E+F	Dual
1	Tetrahedron	4	6	4	2	Self
2	Cube	8	12	6	2	Octahedron
3	Octahedron	6	12	8	2	Cube
4	Icosahedron	12	30	20	2	Dodecahedron
5	Dodecahedron	20	30	12	2	Icosahedron