INTRODUCTION TO EUCLIDS GEOMETRY

INTRODUCTION TO EUCLIDS GEOMETRY

SOME IMPORTANT POINTS TO REMEMBER

·         Though Euclid defined a point, a line and a plane , the definitions are not accepted by mathematicians. Therefore, they are now taken as undefined terms in geometry.

·         Axioms or postulates are the assumptions which are obvious universal truths. They are not proved.

·         Theorems are the assumptions which are proved, using definitions, axioms, previously proved statements and the deductive logic.

·         Some of Euclid's axioms were:

1.      Things which are equal to the same thing are equal to one another.

2.      If equals are added to equals, the wholes are equal.

3.      If equals are subtracted from equals, the remainders are equals.

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

·         Euclid's postulates were:

Postulate 1:  A straight line may be drawn from any one point to any other point.

Postulate 2:  A terminated line can be produced indefinitely.

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

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

Postulate  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 angles are less than two right angles.

·         Two equivalent versions of the Euclid 's 5th postulate are:

                                i.            ' For ever  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 Euclid's Fifth Postulate using the first 4 postulates failed. But they led to the discovery of several other geometries, called  non Euclidean geometrics