non-Euclidean geometry

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                NON-EUCLIDEAN GEOMETRY

                 Every mathematics student to professor everyone heard the name of Euclid. He is famous for his contribution to the mathematics. One of his great contribution is Euclidean geometry. That we study whole our school life, that all based  on Euclid postulate and axioms that he write on his famous book "The Elements", that book has13 chapters based on his geometry.

    And the great thing you know before 19th century mathematicians believe that Euclid geometry is god's word so no another geometry not exist, and but some mathematicians are believe some of Euclid postulate is can derive from other, so some are trying to prove but many tries and fail.

     In 19th century János Bolyai and Nikolai Ivanovich Lobachevsky (separately) thought that instead of proving one of them imply from another we can make or study geometry that not follow some of Euclid postulates .

 Before go on into non-Euclidean geometry we first see Euclid's postulates. 


Euclid’s Postulate: -

1) A straight line segment can be drawn joining any two points.

2) Any straight line segment can be extended indefinitely in a straight line.

3) Given any straight line segment, a circle can be drawn having the segments as radius and one endpoint as center.

4) All right angles are congruent.

5) If two lines are drawn which intersect third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.


Consequence of 5th postulate: -

1) Sum of inner angles in a triangle is 180°

2) There is similar triangle which are not congruent.


One thing you may be noticed that whenever we study geometry, we draw all the pictures in black board (a plane), so obviously all the Euclid’s postulate is true. It’s not Bolayi, Lobachevsky first introduces non-Euclidean geometry a long-time mathematicians study this kind of geometry without knowing that this geometry follows different

Examples- Two special kind of non-Euclidean geometry is Spherical geometry and Hyperbolic geometry.

Spherical geometry: -

I. In a sphere line are great circles (such as the equator or the meridians on a globe), so if the two points are totally opposite to each other then then we can draw many lines joining two points.

II. Euclid’s 2nd postulate holds in spherical geometry.





III. If I give long bigger radius, it’s impossible to draw a circle from this.

IV. Euclid’s 4th postulate holds in spherical geometry.


V. In sphere sum of inner angle of a triangle is not 180°

Hence Euclid’s 1st, 3rd, 5th postulate are violate in spherical geometry. It’s an example of non-Euclidean geometry.

We discuss briefly about hyperbolic geometry later blogs.

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