Introduction To Euclidean Geometry

The word geometry is derived from the ancient Greek terms ‘geo’ which means earth and ‘metron’ meaning measurement. It is a branch of mathematics dealing with shape, size, position, spatial relationships and properties of figures. Geometry, since its origin was studied in many ancient civilizations like Egypt, Babylonia, India, Greece etc. People in these civilizations encountered many practical problems which led to the development of Geometry (Euclidean Geometry).

History of Euclidean 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’. Further the ‘Elements’ was divided into thirteen books which popularized geometry all over the world. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. It was through his works, we have a collective source for learning geometry; Euclidean Geometry lays the foundation for geometry as we know now.

Euclid’s Elements is a mathematical and geometrical work consisting of 13 books written by ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt. Book 1 – 4th and 6th discuss plane geometry. Euclid gave five postulates for plane geometry known as Euclid’s Postulates. The geometry is known as Euclidean geometry. Before discussing Euclid’s Postulates let us discuss a few terms as listed by Euclid in his book 1 of the ‘Elements’.

Euclidean geometry and its postulates:

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, surface has 2, line has 1 and point is dimensionless. A point is anything that  has no part, a breadth-less length is a line and the ends of a line are points. A surface is something which has length and breadth only. It can be seen that the definition of few terms need extra specification. Now let us discuss Euclid’s Postulates in detail.

Euclid’s Postulate 1: A straight line can be drawn from any one 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

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 2

Euclid’s Postulate 3: A circle can be drawn with any centre and any radius.

Euclid’s Postulate 4: All right angles are equal to one another.

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 the two right angles.

Further Euclid’s Postulates and axioms were used by him to prove other geometrical concepts using deductive reasoning. No doubt the foundation of present day geometry was laid by Euclid and his book the ‘Elements’.

So here we had a detailed discussion about euclidean geometry and postulates. We at Byju’s are of the opinion that right guidance with practicing with sample questions and answers are key to ensure success like the ones given here- NCERT solutions for introduction to euclids geometry.

Practise This Question

Two obtuse angles can be supplementary.