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Introduction to Euclid's Geometry

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Introduction to Euclid's Geometry - Lesson Summary

The word 'Geometry' is derived from the Greek words 'Geo' means 'Earth' and 'Metron' means to 'Measure'. Around 325 BC Euclid, a teacher of mathematics at Alexandria in Egypt, collected all the known work and arranged it in his famous treatise, called 'Elements'. He divided the 'Elements' into thirteen chapters, each called a book. These books influenced the whole world's understanding of geometry for generations to come. Euclid listed 23 definitions in book 1 of the 'Elements'.

Few of the Euclid's Definitions:

1. A point is that which has no part.
2. A line is breadthless length.
3. The ends of a line are points.
4. A straight line is a line which lies evenly with the points on itself.
5. A surface is that which has length and breadth only.
6. The edges of a surface are lines.
7. A plane surface is a surface which lies evenly with the straight lines on itself.

The definitions of a point, a line, and a plane,are not accepted by mathematicians. Therefore, these terms are taken as undefined.

Axioms or postulates are the assumptions which are obvious universal truths. They are not proved.
An axiom is a statement which is accepted to be true without proof.
The assumptions that are specific to geometry are called postulates.

Euclid's Postulates:

1. A straight line can be drawn from any point to any point.
2. A terminated line can be produced indefinitely.
3. It is possible to describe a circle with any centre any distance.
4. All right angles are equal to one another.
5. If a straight line falling 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.

Two equivalent versions of the Ffifth Euclid’s postulates:

(i) ‘For every line l and for every point P not lying on l, there exists a unique line m passing through P and parallel to l’.
(ii) Two distinct intersecting lines cannot be parallel to the same line.

All the attempts to prove the fifth postulate of Euclid using the first four postulates and the other axioms as theorem failed. But those efforts led to the creation of several other geometries called the non-Euclidean geometries

Euclid's Axioms:

1. Things which are equal to the same things are also equal to one another.
2. If equals are added to equals, then the wholes are equal.
3. If equals are subtracted from equals, then the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
6. Things which are double of the same things are equal to one another.
7. Things which are halves of the same things are equal to one another.

Theorems are mathematical statements which are proved using definitions, axioms and already proved statements and deductive reasining.


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