Tag Archives: non-euclidean geometry

The Copernicus of Geometry


     A few weeks back we had discussed the method of mathematical proof called “Reductio Ad Absurdum”. We had also learnt that Euclid used this method in several proofs. In that VERITAS I had mentioned that Euclid made a wrong proof of the fifth postulate or the parallel postulate.

        In today’s VERITAS we will discuss the parallel postulate and also read the story of the person who dared to challenge this postulate.

My inspiration for this VERITAS is E.T Bell’s book “Men of Mathematics”.

I can safely bet that anyone who reads this book cannot escape from falling in love with mathematics and a desire to become one of them!

      The parallel postulate can be stated in various ways. The simplest definition is this: ” if there is a straight line and a point X not on that line then there is only one line that can pass through X and never intersect the given line( even if they are both extended to infinity).

This line is parallel to the given line”. This looks pretty obvious if you draw lines on a piece of paper. Euclid tried to prove this but was never really satisfied. So he used it as an axiom. That was about 300 BC. No one challenged this for about 2200 years. And then came Lobatchewsky, the Copernicus of geometry ….

       Nicolas Lobatchewsky was the son of a minor government official. He was born on November 2, 1793 in Russia. He did his schooling at Kazan. He entered the Kazan University in 1807. In 1811 he obtained his degree from the university. In 1813 he was appointed to a teaching post in that University.

      The Kazan University was a new and a small one. And it was in a small town. Lobatchewsky professorship in the university was not restricted to teaching mathematics. There was little order in the university and in the town. So Lobatchewsky did nearly everything : he taught Mathematics, Physics, Astronomy. He was inchange of the laboratories, he was inchange of the library, he had to supervise the elementary schools in the area, he would even dust the books and catalog the books in the library himself. So he did just about everything.

In 1827 Lobatchewsky was appointed the rector of the university.

That was one of the top posts there. But that did not deter Lobatchewsky from doing everything. He would still take a mop and clean the lab.

When the government decided to modernize the university by making a few new buildings, Lobatchewsky was incharge of overseeing the project. But Lobatchewsky could not just oversee! He had to be in it! So he learnt architecture and designed the buildings himself! The university was his life!

         But mathematics was his first love. In 1823 he produced a work that would challenge the Great Euclid himself. He created a new geometry in which the parallel postulate is not valid. And this new geometry was fully self consistent( or complete). This geometry later came to be known as non-Euclidian geometry. Euclid’s geometry was the geometry of flat surfaces, Lobatchewsky’s geometry was the geometry of curved surfaces.

      To get an idea of geometries other than the euclidian lets take an example of our own curved earth. When we travel between 2 distant places the shortest path is not a straight line but a curve. ( Remember the last time you flew from Jalandhar to Southall the TV in the aeroplane showed you a curved path). The shortest distance between two points on the earth is called a geodesic. How is this defined: Imagine a flat plane that passes through the 2 points and cuts the earth and passes through the centre of the earth. The line that this plane makes on the surface of the earth is called a geodesic.

      The geometry on a sphere is different from euclidian geometry. On a plane surface ( euclidian geometry) two lines will either be parallel, or cross each other at exactly one point. On a sphere 2 geodesics will either be parallel or cross each other at TWO points. There are several other differences.

      On a sphere the sum of the angles of a triangle is always more than 180 degrees.

      And there are several other geometries: the hyperbolic geometry and the geometry on a pseudo-sphere. On a pseudo sphere there are exactly 2 lines which pass through a point and are parallel to a line. Then there is Riemann’s geometry in which 2 angles of a rectange are obtuse. Riemann’s geometry was used by Einstein in his General theory of Relativity in which he showed that gravity curves space-time.

      So Lobatchewsky’s genuis was in challenging a 2200 year old axiom. He opened the gates for several types of geometries. He showed that Euclid’s geometry was only a special case.

        But his genuis as a mathematician was never recognized in his lifetime. In 1846 he was charged with mismanagement in the university and was asked to leave. He loved the university, it was HIS university.

This broke him. His health suffered and he became blind. In 1856 he died a sad and broken man.

      Physicists are now realizing that all of Physics and all the laws that govern the Universe may be manifestations of the Geometry of space time.

      A few poetic words on geometry by Emily Dickenson:


      Best Witchcraft is Geometry



      Best Witchcraft is Geometry

        To the magician’s mind —

        His ordinary acts are feats

        To thinking of mankind.





  Go, wondrous creature! mount where Science guides:                 

  Go, measure earth, weigh air, and state the tides:                 

  Instruct the planets in what orbs to run,                           

  Correct old time and regulate the Sun;